École d'Été de Probabilités de Saint-Flour **Lecture Notes in Mathematics 2243**

# Asaf Nachmias

# Planar Maps, Random Walks and Circle Packing

École d'Été de Probabilités de Saint-Flour XLVIII - 2018

# **Lecture Notes in Mathematics** 2243

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# **Saint-Flour Probability Summer School**

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The duration of each school is 12 days (it was 17 days up to 2005), and up to 100 participants can attend it. The aim is to provide, in three high-level courses, a comprehensive study of some fields in probability theory or Statistics. The lecturers are chosen by an international scientific board. The participants themselves also have the opportunity to give short lectures about their research work.

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#### **Christophe Bahadoran**

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**Arnaud Guillin**  Arnaud@uca.fr

**Hacène Djellout** 

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stflour@math.univ-bpclermont.fr

Université Clermont Auvergne - Aubière cedex, France

Asaf Nachmias

# Planar Maps, Random Walks and Circle Packing

École d'Été de Probabilités de Saint-Flour XLVIII - 2018

Asaf Nachmias Department of Mathematical Sciences Tel Aviv University Tel Aviv, Israel

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISSN 0721-5363 École d'Été de Probabilités de Saint-Flour ISBN 978-3-030-27967-7 ISBN 978-3-030-27968-4 (eBook) https://doi.org/10.1007/978-3-030-27968-4

Mathematics Subject Classification (2010): Primary: 82B41; Secondary: 52C26

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## **Preface**

These lecture notes are intended to accompany a single-semester graduate course. They are meant to be *entirely self-contained*. All the theory required to prove the main results is presented and only basic knowledge in probability theory is assumed.

In Chap. 1, we describe the main storyline of this text. It is meant to be light bedtime reading exposing the reader to the main results that will be presented and providing some background. Chapter 2 introduces the theory of electric networks and discusses their highly useful relations to random walks. It is roughly based on Chap. 8 of Yuval Peres' excellent lecture notes [69]. We then discuss the circle packing theorem and present its proof in Chap. 3. Chapter 4 discusses the beautiful theorem of He and Schramm [40], relating the circle packing type of a graph to recurrence and transience of the random walk on it. To the best of our knowledge, their work is the first to form connections between the circle packing theorem and probability theory. Next in Chap. 5, we present the highly influential theorem of Benjamini and Schramm [11] about the almost sure recurrence of the simple random walk in planar graph limits of bounded degrees. The notion of a *local limit* (also known as *distributional limit* or *Benjamini-Schramm limit*) of a sequence of finite graphs was introduced there for the first time to our knowledge (and also studied by Aldous–Steele [3] and Aldous–Lyons [2]); this notion is highly important in probability theory as well as other mathematical disciplines (see [2] and the references within). In Chap. 6, we provide a theorem from which one can deduce the almost sure recurrence of the simple random walk on many models of random planar maps. This theorem was obtained by Ori Gurel-Gurevich and Nachmias in [31]. Chapter 7 discusses uniform spanning forests on planar maps and appeals to the circle packing theorem to show that the free uniform spanning forest on proper planar maps is almost surely connected, i.e., it is in fact a tree. This theorem was obtained by Hutchcroft and Nachmias in [45]. We close these notes in Chap. 8 with a description of some related contemporary developments in this field that are not presented in this text.

We have made an effort to add value beyond what is in the published papers. Our proof of the circle packing theorem in Chap. 3 is inspired by Thurston's argument [82] and Brightwell–Scheinerman [13], but we have made what we think are some simplifications; the proof also employs a neat argument due to Ohad Feldheim and Ori Gurel-Gurevich (Theorem 3.14) which makes the drawing part of the argument rather straightforward and avoids topological considerations that are used in the classical proofs. The original proof of the He–Schramm Theorem [40] is based on the notion of *discrete extremal length* which is essentially a form of *effective resistance* in electric networks (in fact, the *edge* extremal length is precisely an effective resistance, see [61, Exercise 2.78]). We find that our approach in Chap. 4 using electric networks is somewhat more robust and intuitive to probabilists. We obtain a quantitative version of the He-Schramm Theorem in Chap. 4 as well as the Benjamini–Schramm Theorem [11] in Chap. 5 (see Theorem 5.8). These quantified versions are key to the proofs of Chap. 6. Lastly, some aspects of stationary random graphs are better explained here in Chap. 6 than in the publication [31]. The videoed lectures of this course taken in 48th Saint-Flour summer school are available at the author's webpage http://www.math.tau.ac.il/~asafnach/.

#### **Acknowledgments**

I would like to deeply thank Daniel Jerison, Peleg Michaeli, and Matan Shalev for typing, editing, and proofreading most of this text and for the many comments, corrections, and suggestions. I am indebted to Tom Hutchcroft for his assistance in writing the introduction and for surveying related topics not included in these notes (Chaps. 1 and 8). I thank Sébastien Martineau, Pierre Petit, Dominik Schmid, and Mateo Wirth for their corrections and comments to this text. I am also grateful to the participants of the 48th Saint-Flour Summer School and its organizers, Christophe Bahadoran, Arnaud Guillin, and Hacène Djellout, for a very enjoyable summer school.

I am beholden to my collaborators on these topics: Omer Angel, Martin Barlow, Itai Benjamini, Nicolas Curien, Ori Gurel-Gurevich, Tom Hutchcroft, Daniel Jerison, Gourab Ray, Steffen Rohde, and Juan Souto. I have learned a lot from our work and conversations. Special thanks go to Ori Gurel-Gurevich for embarking together on this research endeavor beginning in 2011 at the University of British Columbia, Vancouver, Canada. Many of the ideas and methods presented in these notes were obtained in our joint work.

I am also highly indebted to the late Oded Schramm whose mathematical work, originality, and vision, especially in the topics studied in these notes, have been an enormous source of inspiration. It is no coincidence that his name appears on almost every other page here. It has become routine for my collaborators and I to ask ourselves "What would Oded do?" hoping that reflecting on this question would cut right to the heart of matters. Steffen Rohde's wonderful survey [71] of Oded's work is very much recommended.

Lastly, I thank Shira Wilkof and baby Ada for muse and inspiration.

July 2019

Tel Aviv, Israel Asaf Nachmias1

<sup>1</sup>This project has received funding from the European Research Council (ERC) under the European Union's, Horizon 2020 research and innovation programme (grant agreement number 676970, RANDGEOM).

# **Contents**



# **Chapter 1 Introduction**

#### **1.1 The Circle Packing Theorem**

A planar graph is a graph that can be drawn in the plane, with vertices represented by points and edges represented by non-crossing curves. There are many different ways of drawing any given planar graph and it is not clear what is a canonical method. One very useful and widely applicable method of drawing a planar graph is given by Koebe's 1936 *circle packing theorem* [51], stated below. As we will see, various geometric properties of the circle packing drawing (such as existence of accumulation points and their structure, bounds on the radii of circles and so on) encode important probabilistic information (such as the recurrence/transience of the simple random walk, connectivity of the uniform spanning forest and much more). This deep connection is especially fruitful to the study of random planar maps. Indeed, one of the main goals of these notes is to present a self-contained proof that the so-called *uniform infinite planar triangulation* (UIPT) is almost surely recurrent [31].

<sup>A</sup> **circle packing** is a collection of discs <sup>P</sup> = {Cv} in the plane <sup>C</sup> such that any two distinct discs in P have disjoint interiors. That is, distinct discs in P may be tangent, but may not overlap. Given a circle packing P, we define the **tangency graph** G(P ) of P to be the graph with vertex set P and with two vertices connected by an edge if and only if their corresponding circles are tangent. The tangency graph G(P ) can be drawn in the plane by drawing straight lines between the centers of tangent circles in P, and is therefore planar. It is also clear from the definition that G(P ) is **simple**, that is, any two vertices are connected by at most one edge and there are no edges beginning and ending at the same vertex. See Fig. 1.1.

We call a circle packing P a circle packing of a planar graph G if G(P ) is isomorphic to G.

**Theorem 1.1 (Koebe 1936)** *Every finite simple planar graph* G *has a circle packing. That is, there exists a circle packing* P *such that* G(P ) *is isomorphic to* G*.*

A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes

in Mathematics 2243, https://doi.org/10.1007/978-3-030-27968-4\_1

**Fig. 1.1** A planar graph and a circle packing of it

**Fig. 1.2** A sketch of how to obtain circle packings using Koebe's extension of the Riemann mapping theorem to finitely connected domains, which states that every domain <sup>D</sup> <sup>⊆</sup> <sup>C</sup> ∪ {∞} with at most finitely many boundary components is conformally equivalent to a *circle domain*, that is, a domain all of whose boundary components are circles or points. (**a**) Step 1: We begin by drawing the finite simple planar graph G in the plane in an arbitrary way. (**b**) Step 2: If we remove the 'middle ε' of each edge, then the complement of the resulting drawing is a domain with finitely many boundary components. (**c**) Step 3: Finding a conformal map from this domain to a circle domain gives an 'approximate circle packing' of G. (**d**) Step 4: Taking the limit as ε ↓ 0 can be proven to yield a circle packing of G

One immediate consequence of the circle packing theorem is Fáry's Theorem [25], which states that every finite simple planar graph can be drawn so that all the edges are represented by straight lines.

The circle packing theorem was first discovered by Koebe [51], who established it as a corollary to his work on the generalization of the Riemann mapping theorem to finitely connected domains; a brief sketch of Koebe's argument is given in Fig. 1.2. The theorem was rediscovered and popularized in the 1970s by Thurston [82], who showed that it follows as a corollary to the work of Andreev on hyperbolic polyhedra (see also [63]). Thurston also initiated a popular program of understanding circle packing as a form of *discrete complex analysis*, a viewpoint which has been highly influential in the subsequent development of the subject and which we discuss in more detail below (see [79] for a review of a different form of discrete complex analysis with many applications to probability). There are now many proofs of the circle packing theorem available including, remarkably, four distinct proofs discovered by Oded Schramm. In Chap. 3 we will give an entirely combinatorial proof, which is adapted from the proof of Thurston [63, 82] and Brightwell and Scheinerman [13].

#### *Uniqueness*

We cannot expect a uniqueness statement in Theorem 1.1 (see Fig. 1.1; we may "slide" circles 5 and 6 along circle 2). However, when our graph is a *finite triangulation*, circle packings enjoy uniqueness up to circle-preserving transformations.

**Definition 1.2** A planar **triangulation** is a planar graph that can be drawn so that every face is incident to exactly three edges. In particular, when the graph is finite this property must hold for the outer face as well.

*Claim 1.3* If G is a finite triangulation, then the circle packing whose tangency graph is isomorphic to G is unique, up to Möbius transformations and reflections in lines.

The uniqueness of circle packing was first proven by Thurston, who noted that it follows as a corollary to Mostow's rigidity theorem. Since then, many different proofs have been found. In Chap. 3 we will give a very short and elementary proof of uniqueness due to Oded Schramm that is based on the maximum principle.

#### *Infinite Planar Graphs*

So far, we have only discussed the existence and uniqueness of circle packings of *finite* planar triangulations. What happens with infinite triangulations? To address this question, we will need to introduce some more definitions.

**Definition 1.4** We say that a graph G is **one-ended** if the removal of any finite set of vertices leaves at most one infinite connected component.

**Definition 1.5** Let P = {Cv} be a circle packing of a triangulation. We define the **carrier** of P to be the union of the closed discs bounded by the circles of P together with the spaces bounded between any three circles that form a face (i.e., the interstices). We say that P is **in** D if its carrier is D.

See Fig. 1.3 for examples where the carrier is a disc or a square. The circle packing of the standard triangular lattice (see Fig. 4.2) has the whole plane C as its carrier. It is not too hard to see that if G(P ) is an infinite triangulation, then it is one-ended if and only if the carrier of P is simply connected, see Lemma 4.1.

It can be shown via a compactness argument that any simple infinite planar triangulation can be circle packed in *some* domain. Indeed, one can simply take subsequential limits of circle packings of finite subgraphs (the fact that such subsequential limits can be taken is a consequence of the so-called Ring Lemma, see Lemma 4.2). This is performed in Claim 4.3. However, this compactness argument does not give us any control of the domain we end up with as the carrier of our circle packing. The following theorems of He and Schramm [39, 40] give us much

**Fig. 1.3** The 7-regular hyperbolic tessellation circle packed in a disc and in a square

better control; they can be thought of as discrete analogues of the Poincaré-Koebe uniformization theorem for Riemann surfaces.

**Theorem 1.6 (He and Schramm 1993)** *Any one-ended infinite triangulation can be circle packed such that the carrier is either the plane or the open unit disk, but not both.*

This theorem will be proved in Chap. 4 (with the added assumption of finite maximal degree). The proofs in [39, 40] are based on the notion of *discrete extremal length*. We will present our own approach to the proof in Chap. 4 based on a very similar notion of electric resistance discussed in Chap. 2. This approach is somewhat more appealing to a probabilist and allows for quantitative versions of the He-Schramm Theorem that will be used later for the study of random planar maps in Chap. 6.

In view of Theorem 1.6, we call an infinite one-ended simple planar triangulation **CP parabolic** if it can be circle packed in C, and call it **CP hyperbolic** if it can be circle packed in the open unit disk U.

**Theorem 1.7 (He and Schramm 1995)** *Let* T *be a CP hyperbolic infinite oneended simple planar triangulation and let* D - C *be a simply connected domain. Then there exists a circle packing of* T *with carrier* D*.*

What about uniqueness? Theorem 1.7 shows that, in general, we have much more flexibility when choosing a circle packing of an infinite planar triangulation than we have in the finite case, see Fig. 1.3 again. Indeed, it implies that the circle packing of a CP hyperbolic triangulation is *not* determined up to Möbius transformations and reflections, since, for example, we can circle pack the same triangulation in both the unit disc and the unit square, and these two packings are clearly not related by a Möbius transformation. Fortunately, the following theorem of Schramm [73] shows that we recover Möbius rigidity if we restrict the packing to be in C or U.

**Theorem 1.8 (Schramm 1991)** *Let* T *be a one-ended infinite planar triangulation.*


#### *Relation to Conformal Mapping*

A central motivation behind Thurston's popularization of circle packing was its role as a discrete analogue of conformal mapping. The resulting theory is somewhat tangential to the main thrust of these notes, but is worth reviewing for its beauty, and for the intuition it gives about circle packing. A more detailed treatment of this and related topics is given in [81].

Recall that a map <sup>φ</sup> : <sup>D</sup> <sup>→</sup> <sup>D</sup> between two domains D,D <sup>⊆</sup> <sup>C</sup> is conformal if and only if it is holomorphic and one-to-one. Intuitively, we can think of the latter condition as saying that φ maps infinitesimal circles to infinitesimal circles. Thus, it is natural to wonder, as Thurston did, whether conformal maps can be approximated by graph isomorphisms between circle packings of the corresponding domains, which *literally* map circles to circles.

For each ε > 0, let <sup>T</sup><sup>ε</sup> = {εn <sup>+</sup> <sup>ε</sup> <sup>1</sup><sup>+</sup> <sup>√</sup>3<sup>i</sup> <sup>2</sup> <sup>m</sup> : n, m <sup>∈</sup> <sup>Z</sup>} ⊆ <sup>C</sup> be the triangular lattice with lattice spacing ε, which we make into a simple planar triangulation by connecting two vertices if and only if they have distance ε from each other. This triangulation is naturally circle packed in the plane by placing a disc of radius ε around each point of Tε: this is known as the **hexagonal packing**. Now, let D be a simply connected domain, and take z<sup>0</sup> to be a marked point in the interior of D. For each ε > 0 let uε be an element of <sup>T</sup><sup>ε</sup> of minimal distance to <sup>z</sup>0, and let vε <sup>=</sup> uε <sup>+</sup><sup>ε</sup> and wε <sup>=</sup> uε <sup>+</sup> (<sup>1</sup> <sup>+</sup> <sup>√</sup>3i)ε/2. For each ε > 0, let Tε(D) be the subgraph of <sup>T</sup><sup>ε</sup> induced by the vertices of distance at least 2ε from ∂D (i.e., the subgraph containing all such vertices and all the edges between them), and let T <sup>ε</sup> (D) be the component of Tε(D) containing uε. Finally, let T <sup>ε</sup> (D) be the triangulation obtained from T <sup>ε</sup> (D) by placing a single additional vertex ∂ε in the outer face of T <sup>ε</sup> (D) and connecting this vertex to every vertex in the outer boundary of T <sup>ε</sup> (D).

Applying the circle packing theorem to T <sup>ε</sup> (D) and then applying a Möbius transformation or a reflection if necessary, we obtain a circle packing Pε of T <sup>ε</sup> (D) with the following properties:


The function sending each vertex of T <sup>ε</sup> (D) to the center of the circle representing it in Pε can be extended piecewise on each triangle by an affine extension. Call the resulting function φε.

The following theorem was conjectured by Thurston and proven by Rodin and Sullivan [70].

**Theorem 1.9 (Rodin and Sullivan 1987)** *Let* φ *be the unique conformal map from* <sup>D</sup> *to* <sup>U</sup> *with* φ(z0) <sup>=</sup> <sup>0</sup> *and* <sup>φ</sup> (z0) > 0*. Then* φε *converge to* φ *as* ε ↓ 0*, uniformly on compact subsets of* D*.*

See Fig. 1.4. The key to the proof of Theorem 1.9 was to establish that the hexagonal packing is the only circle packing of the triangular lattice, which is now a special case of Theorem 1.8.

Various strengthenings and generalizations of Theorem 1.9 have been established in the works [21, 36, 38, 41, 42, 80].

**Fig. 1.4** Approximating the conformal map from a rhombus to the disc using circle packing, at two different degrees of accuracy

#### **1.2 Probabilistic Applications**

Why should we be interested in circle packing as probabilists? At a very heuristic level, when we uniformize the *geometry* of a triangulation by applying the circle packing theorem, we also uniformize the *random walk* on the triangulation, allowing us to compare it to a standard reference process that we understand very well, namely Brownian motion. Indeed, since Brownian motion is conformally invariant and circle packings satisfy an approximate version of conformality, it is not unreasonable to expect that the random walk on a circle packed triangulation will behave similarly to Brownian motion. This intuition turns out to be broadly correct, at least when the triangulation has bounded degrees, although it is more accurate to say that the random walk behaves like a *quasi-conformal image* of Brownian motion, that is, the image of Brownian motion under a function that distorts angles by a bounded amount.

Although it is possible to make the discussion in the paragraph above precise, in these notes we will be interested primarily in much coarser information that can be extracted from circle packings, namely *effective resistance estimates* for planar graphs. This fundamental topic is thoroughly discussed in Chap. 2. One of the many definitions of the effective resistance *R*eff(A ↔ B) between two disjoint sets A and B in a finite graph is

$$\frac{1}{\mathcal{R}\_{\text{eff}}(A \leftrightarrow B)} = \sum\_{v \in A} \deg(v) \mathbb{P}\_v(\tau\_B < \tau\_A^+),$$

where P<sup>v</sup> is the law of the simple random walk started at v, τB is the first time the walk hits B, and τ + <sup>A</sup> is the first positive time the walk visits A. Good enough control of effective resistances allows one to understand most aspects of the random walk on a graph. We can also define effective resistances on infinite graphs, although issues arise with boundary conditions. An infinite graph is recurrent if and only if the effective resistance from a vertex to infinity is infinite.

The effective resistance can also be computed via either of two variational principles: *Dirichlet's principle* and *Thomson's principle*, see Sect. 2.4. The first expresses the effective resistance as a *supremum* of energies of a certain set of *functions*, while the second expresses the effective resistance as an *infimum* of energies of a certain set of *flows*. Thus, we can bound effective resistances from above by constructing flows, and from below by constructing functions. A central insight is that we can *use the circle packing* to construct these functions and flows. This idea leads fairly easily to various statements such as the following:

• The effective resistance across a Euclidean annulus of fixed modulus is at most a constant. If the triangulation has bounded degrees, then the resistance is at least a constant.

• The effective resistance between the left and right sides of a Euclidean square is at most a constant. If the triangulation has bounded degrees, then the resistance is at least a constant.

See for instance Lemma 4.9. We will use these ideas to prove the following remarkable theorem of He and Schramm [40], which pioneered the connection between circle packing and random walks.

**Theorem 1.10 (He and Schramm 1995)** *Let* T *be a one-ended infinite triangulation. If* T *has bounded degrees, then it is CP parabolic if and only if it is recurrent for simple random walk, that is, if and only if the simple random walk on* T *visits every vertex infinitely often almost surely.*

This has been extended to the multiply-ended cases in [32], see also Chap. 8, item 4.

#### *Recurrence of Distributional Limits of Random Planar Maps*

Random planar maps is a widely studied field lying at the intersection of probability, combinatorics and statistical physics. It aims to answer the vague question "what does a typical random surface look like?"

We provide here a very quick account of this field, referring the readers to the excellent lecture notes [58] by Le Gall and Miermont, and the many references within for further reading. The enumerative study of planar maps (answering questions of the form "how many simple triangulations on n vertices are there?") began with the work of Tutte in the 1960s [83] who enumerated various classes of finite planar maps, in particular triangulations. Cori and Vauquelin [18], Schaeffer [72] and Chassaing and Schaeffer [16] have found beautiful bijections between planar maps and labeled trees and initiated this fascinating topic in enumerative combinatorics. The bijections themselves are model dependent and extremely useful since many combinatorial and metric aspects of random planar maps can be inferred from them. This approach has spurred a new line of research: limits of large random planar maps.

Two natural notions of such limits come to mind: scaling limits and local limits. In the first notion, one takes a random planar map Mn on n vertices, scales the distances appropriately (in most models the correct scaling turns out to be n−1/4), and aims to show that this random metric space converges in distribution in the Gromov-Hausdorff sense. The existence of such limits was suggested by Chassaing and Schaeffer [16], Le Gall [55], and Marckert and Mokkadem [62], who coined the term *the Brownian map* for such a limit. The recent landmark work of Le Gall [56] and Miermont [65] establishes the convergence of random p-angulations for p = 3 and all even p to the Brownian map.

The study of local limits of random planar maps, initiated by Benjamini and Schramm [11], while bearing many similarities, is independent of the study of scaling limits. The *local* limit of a random planar map Mn on n vertices is an infinite random rooted graph (U, ρ) with the property that neighborhoods of Mn around a random vertex converge in distribution to neighborhoods of U around ρ. The infinite random graph (U, ρ) captures the local behavior of Mn around typical vertices. We develop this notion precisely in Chap. 5.

In their pioneering work, Angel and Schramm [5] showed that the local limit of a uniformly chosen random triangulation on n vertices exists and that it is a oneended infinite planar triangulation. They termed the limit as the *uniform infinite planar triangulation* (UIPT). The uniform infinite planar quadrangulation (UIPQ), that is, the local limit of a uniformly chosen random quadrangulation (i.e., each face has 4 edges) on n vertices, was later constructed by Krikun [52].

The questions in this line of research concern the almost sure properties of this limiting geometry. It is a highly fractal geometry that is drastically different from Z2. Angel [4] proved that the volume of a graph-distance ball of radius r in the UIPT is almost surely of order r4+o(1) and that the boundary component separating this ball from infinity has volume r2+o(1) almost surely. For the UIPQ this is proved in [16].

Due to the various combinatorial techniques of generating random planar maps, many of the metric properties of the UIPT/UIPQ are firmly understood. Surface properties of these maps are somewhat harder to understand using enumerative methods. Recall that a non-compact simply connected Riemannian surface is either conformally equivalent to the disc or the whole plane and that this is determined according to whether Brownian motion on the surface is transient or recurrent. Hence, the behavior of the simple random walk on the UIPT/UIPQ is considered here as a "surface property" (see also [30]).

As mentioned earlier, one of the main objectives of these notes is to answer the question of the almost sure recurrence/transience of the simple random walk on the UIPT/UIPQ. We provide a general statement, Theorem 6.1 of these notes, to which a corollary is

#### **Theorem 1.11 ([31])** *The UIPT and UIPQ are almost surely recurrent.*

The proof heavily relies on the circle packing theorem and can be viewed as an extension of the remarkable theorem of Benjamini and Schramm [11] stating that the local limit of finite planar maps with finite maximum degree is almost surely recurrent. The maximum degree of the UIPT is unbounded and so one cannot apply [11]. A combination of the techniques presented in Chaps. 4–6 is required to overcome this difficulty.

Recently, there have been terrific new developments studying further surface properties of the UIPT/UIPQ. Lee [59] has given an exciting new proof of Theorem 1.11 based on a spectral analysis and an embedding theorem for planar maps due to [48]. His proof also yields that the spectral dimension of the UIPT/UIPQ is at most 2 and applies to local limits of sphere-packable graphs in higher dimensions as well. Gwynne and Miller [33] provided the converse bound showing that the spectral dimension of the UIPT equals 2 and calculated other exponents governing the behavior of the random walk. Their results are based on the deep work of Gwynne et al. [35] (see also Chap. 8, item 9).

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 2 Random Walks and Electric Networks**

An extremely useful tool and viewpoint for the study of random walks is Kirchhoff's theory of electric networks. Our treatment here roughly follows [69, Chapter 8], we also refer the reader to [61] for an in-depth comprehensive study.

**Definition 2.1** A **network** is a connected graph G = (V , E) endowed with positive edge weights, {ce}e∈<sup>E</sup> (called **conductances**). The reciprocals re = 1/ce are called **resistances**.

In Sects. 2.1–2.4 below we discuss finite networks. We extend our treatment to infinite networks in Sect. 2.5.

#### **2.1 Harmonic Functions and Voltages**

Let G = (V , E) be a finite network. In physics classes it is taught that when we impose specific voltages at fixed vertices a and z, then current flows through the network according to certain laws (such as the series and parallel laws). An immediate consequence of these laws is that the function from V to R giving the voltage at each vertex is harmonic at each x ∈ V \ {a,z}.

**Definition 2.2** A function <sup>h</sup> : <sup>V</sup> <sup>→</sup> <sup>R</sup> is **harmonic** at a vertex <sup>x</sup> if

$$h(\mathbf{x}) = \frac{1}{\pi\_{\chi}} \sum\_{\mathbf{y} : \mathbf{y} \sim \chi} c\_{\chi \mathbf{y}} h(\mathbf{y}) \qquad \text{where} \qquad \pi\_{\chi} := \sum\_{\mathbf{y} : \mathbf{y} \sim \chi} c\_{\chi \mathbf{y}}.\tag{2.1}$$

Instead of starting with the physical laws and proving that voltage is harmonic, we now take the axiomatically equivalent approach of defining voltage to be a harmonic function and deriving the laws as corollaries.

**Definition 2.3** Given a network G = (V , E) and two distinct vertices a,z ∈ V , a **voltage** is a function <sup>h</sup> : <sup>V</sup> <sup>→</sup> <sup>R</sup> that is harmonic at any <sup>x</sup> <sup>∈</sup> <sup>V</sup> \ {a,z}.

We will show in Claim 2.8 and Corollary 2.7 that for any α, β <sup>∈</sup> <sup>R</sup>, there is a unique voltage h such that h(a) = α and h(z) = β (this assertion is true only when the network is finite).

*Claim 2.4* If h1, h<sup>2</sup> are harmonic at x then so is any linear combination of h1, h2. *Proof* Let <sup>h</sup>¯ <sup>=</sup> αh<sup>1</sup> <sup>+</sup> βh<sup>2</sup> for some α, β <sup>∈</sup> <sup>R</sup>. It holds that

$$\bar{h}(\mathbf{x}) = \alpha h\_1(\mathbf{x}) + \beta h\_2(\mathbf{x}) = \frac{1}{\pi\_\mathbf{x}} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} c\_{\mathbf{xy}} \alpha h\_1(\mathbf{y}) + \frac{1}{\pi\_\mathbf{x}} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} c\_{\mathbf{xy}} \beta h\_2(\mathbf{y})$$

$$= \frac{1}{\pi\_\mathbf{x}} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} c\_{\mathbf{xy}} \bar{h}(\mathbf{y}). \tag{7}$$

*Claim 2.5* If <sup>h</sup> : <sup>V</sup> <sup>→</sup> <sup>R</sup> is harmonic at all the vertices of a finite network, then it is constant.

*Proof* Let M = sup<sup>x</sup> h(x) be the maximum value of h. Let A = {x ∈ V : h(x) = M}. Since G is finite, A = ∅. Given x ∈ A, we have that h(y) ≤ h(x) for all neighbors y of x. By harmonicity, h(x) is the weighted average of the values of h(y) at the neighbors; but this can only happen if all neighbors of x are also in A. Since G is connected we obtain that A = V implying that h is constant. 

We now show that a voltage is determined by its boundary values, i.e., by its values at a,z.

*Claim 2.6* If h is a voltage satisfying h(a) = h(z) = 0, then h ≡ 0.

*Proof* Put M = max<sup>x</sup> h(x) (which is attained since G is finite) and let A = {x ∈ V : h(x) = M}. As before, by harmonicity, if x ∈ A\ {a,z} then all of its neighbors are also in A. Since G is connected, there exists a simple path from x to either a or z such that only its endpoint is in {a,z}. Since h(a) = h(z) = 0 we learn that M = 0, that is, h is non-positive. Similarly, one proves that h is non-negative, thus h ≡ 0. 

**Corollary 2.7 (Voltage Uniqueness)** *For every* α, β <sup>∈</sup> <sup>R</sup>*, if* h, h *are voltages satisfying* h(a) = h (a) = α *and* h(z) = h (z) = β*, then* h ≡ h *.*

*Proof* By Claim 2.4, the function h − h is a voltage, taking the value 0 at a and z, hence by Claim 2.6 we get h ≡ h . 

*Claim 2.8* For every α, β <sup>∈</sup> <sup>R</sup>, there exists a voltage <sup>h</sup> satisfying h(a) <sup>=</sup> <sup>α</sup>,h(z) = β.

*Proof 1* We write n = |V |. Observe that a voltage h with h(a) = α and h(z) = β is defined by a system of n − 2 linear equations of the form (2.1) in n − 2 variables (which are the values h(x) for x ∈ V \ {a,z}). Corollary 2.7 guarantees that the matrix representing that system has empty kernel, hence it is invertible. 

We present an alternative proof of existence based on the random walk on the network. Consider the Markov chain {Xn} on the state space V with transition probabilities

$$p\_{\mathbf{x}\mathbf{y}} := \mathbb{P}(X\_{\mathbf{I}+\mathbf{l}}=\mathbf{y} \mid X\_{\mathbf{l}}=\mathbf{x}) = \frac{c\_{\mathbf{x}\mathbf{y}}}{\pi\_{\mathbf{x}}}.\tag{2.2}$$

This Markov chain is a **weighted random walk** (note that if cxy are all 1 then the described chain is the so-called **simple random walk**). We write P<sup>x</sup> and E<sup>x</sup> for the probability and expectation, respectively, conditioned on X<sup>0</sup> = x. For a vertex x, define the **hitting time** of x by

$$
\pi\_{\boldsymbol{\chi}} := \min \{ t \ge 0 \mid X\_t = \boldsymbol{x} \}.
$$

*Proof 2* We will find a voltage g satisfying g(a) = 0 and g(z) = 1 by setting

$$\mathbf{g}(\mathbf{x}) = \mathbb{P}\_{\mathbf{x}}(\mathfrak{r}\_{\mathfrak{z}} < \mathfrak{r}\_{a}).$$

Indeed, g is harmonic at x = a,z, since by the law of total probability and the Markov property we have

$$\begin{split} \mathbf{g}(\mathbf{x}) = \frac{1}{\pi\_{\mathbf{x}}} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} c\_{\mathbf{xy}} \mathbb{P}\_{\mathbf{x}} (\mathbf{z}\_{\mathbf{z}} < \mathbf{z}\_{a} \mid X\_{\mathbf{i}} = \mathbf{y}) = \frac{1}{\pi\_{\mathbf{x}}} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} c\_{\mathbf{xy}} \mathbb{P}\_{\mathbf{y}} (\mathbf{z}\_{\mathbf{z}} < \mathbf{z}\_{a}) \\ &= \frac{1}{\pi\_{\mathbf{x}}} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} c\_{\mathbf{xy}} \mathbf{g}(\mathbf{y}). \end{split}$$

For general boundary conditions α, β we define h by

$$h(\mathbf{x}) = \mathbf{g}(\mathbf{x}) \cdot (\boldsymbol{\beta} - \boldsymbol{\alpha}) + \boldsymbol{\alpha} \ .$$

By Claim 2.4, h is a voltage, and clearly h(a) = α and h(z) = β, concluding the proof. 

This proof justifies the equality between simple random walk probabilities and voltages that was discussed at the start of this chapter: since the function x → <sup>P</sup>x(τz < τa) is harmonic on <sup>V</sup> \ {a,z} and takes values 0, 1 at a,z respectively, it must be equal to the voltage at x when voltages 0, 1 are imposed at a,z.

*Claim 2.9* If h is a voltage with h(a) ≤ h(z), then h(a) ≤ h(x) ≤ h(z) for all x ∈ V .

Furthermore, if h(a) < h(z) and x ∈ V \ {a,z} is a vertex such that x is in the connected component of z in the graph G\ {a}, and x is in the connected component of a in the graph G \ {z}, then h(a) < h(x) < h(z).

*Proof* This follows directly from the construction of h in Proof 2 of Claim 2.8 and the uniqueness statement of Corollary 2.7. Alternatively, one can argue as in the proof of Claim 2.6 that if M = max<sup>x</sup> h(x) and m = min<sup>x</sup> h(x), then the sets A = {x ∈ V : h(x) = M} and B = {x ∈ V : h(x) = m} must each contain at least one element of {a,z}.

To prove the second assertion, we note that by Claim 2.8 and Corollary 2.7 it is enough to check when h is the voltage with boundary values h(a) = 0 and h(z) = 1. In this case, the condition on x guarantees that the probabilities that the random walk started at x visits a before z or visits z before a are positive. By proof 2 of Claim 2.8 we find that h(x) ∈ (0, 1). 

#### **2.2 Flows and Currents**

For a graph G = (V , E), denote by E the set of edges of G, each endowed with the two possible orientations. That is, (x, y) ∈ E iff {x,y} ∈ E (and in that case, (y, x) ∈ E as well).

**Definition 2.10** <sup>A</sup> **flow from** <sup>a</sup> **to** <sup>z</sup> in a network <sup>G</sup> is a function <sup>θ</sup> : <sup>E</sup> <sup>→</sup> <sup>R</sup> satisfying


*Claim 2.11* If θ1, θ<sup>2</sup> are flows then, so is any linear combination of θ1, θ2.

*Proof* Let <sup>θ</sup>¯ <sup>=</sup> αθ<sup>1</sup> <sup>+</sup> βθ<sup>2</sup> for some α, β <sup>∈</sup> <sup>R</sup>. It holds that

$$
\theta(\mathbf{x}\mathbf{y}) = a\theta\_1(\mathbf{x}\mathbf{y}) + \beta\theta\_2(\mathbf{x}\mathbf{y}) = -a\theta\_1(\mathbf{y}\mathbf{x}) - \beta\theta\_2(\mathbf{y}\mathbf{x}) = -\theta(\mathbf{y}\mathbf{x}),
$$

and for x = a,z,

$$\sum\_{\mathbf{y}:\mathbf{y}\sim\mathbf{x}} \bar{\theta}(\mathbf{x}\mathbf{y}) = \alpha \sum\_{\mathbf{y}:\mathbf{y}\sim\mathbf{x}} \theta\_1(\mathbf{x}\mathbf{y}) + \beta \sum\_{\mathbf{y}:\mathbf{y}\sim\mathbf{x}} \theta\_2(\mathbf{x}\mathbf{y}) = 0. \tag{7}$$

**Definition 2.12** Given a voltage h, the **current flow** θ = θh associated with h is defined by θ (xy) = cxy(h(y) − h(x)).

In other words, the voltage difference across an edge is the product of the current flowing along the edge with the resistance of the edge. This is known as **Ohm's law**. According to this definition, the current flows from vertices with lower voltage to vertices with higher voltage. We will use this convention throughout, but the reader should be advised that some other sources use the opposite convention.

*Claim 2.13* The current flow associated with a voltage is indeed a flow.

*Proof* The current flow is clearly antisymmetric by definition. To show that it satisfies the node law, observe that for x = a,z, since h is harmonic,

$$\sum\_{\mathbf{y}:\mathbf{y}\sim\mathbf{x}}\theta(\mathbf{x}\mathbf{y}) = \overbrace{\sum\_{\mathbf{y}:\mathbf{y}\sim\mathbf{x}}c\_{\mathbf{xy}}h(\mathbf{y})}^{=\pi\_{\mathbf{x}}h(\mathbf{x})} - \overbrace{\sum\_{\mathbf{y}:\mathbf{y}\sim\mathbf{x}}c\_{\mathbf{xy}}h(\mathbf{x})}^{=\pi\_{\mathbf{x}}h(\mathbf{x})} = 0.\tag{7}$$

*Claim 2.14* The current flow associated with a voltage h satisfies **Kirchhoff's cycle law**, that is, for every directed cycle e1,..., em,

$$\sum\_{i=1}^r r\_{e\_i} \theta(\vec{e}\_i) = 0.$$

*Proof* Write e<sup>i</sup> = (xi−<sup>1</sup>, xi), and observe that x<sup>0</sup> = xm. We have that

$$\sum\_{l=1}^{m} r\_{\mathbf{e}\_l} \theta(\vec{e}\_l) = \sum\_{l=1}^{m} r\_{\mathbf{x}\_{l-1}\mathbf{x}\_l} c\_{\mathbf{x}\_{l-1}\mathbf{x}\_l} (h(\mathbf{x}\_l) - h(\mathbf{x}\_{l-1})) = \sum\_{l=1}^{m} (h(\mathbf{x}\_l) - h(\mathbf{x}\_{l-1})) = 0.$$

For examples of a flow which does not satisfy the cycle law and a current flow, see Fig. 2.1.

*Claim 2.15* Given a flow θ which satisfies the cycle law, there exists a voltage h = hθ such that θ is the current flow associated with h. Furthermore, this voltage is unique up to an additive constant.

*Proof* For every vertex x, let e1,..., e<sup>k</sup> be a path from a to x, and define

$$h(\mathbf{x}) = \sum\_{l=1}^{k} r\_{\ell l} \theta(\vec{e}\_l). \tag{2.3}$$

**Fig. 2.1** On the left, a flow of strength 2 in which the cycle law is violated. On the right, the unit (i.e., strength 1) current flow

Note that since θ satisfies the cycle law, the right hand side of (2.3) does not depend on the choice of the path, hence h(x)is well defined. Let x ∈ V , and consider a given path e1,..., e<sup>k</sup> from a to x (if x = a we take the empty path). To evaluate h(y) for y ∼ x, consider the path e1,..., ek, xy from a to y, so h(y) = h(x) + rxyθ (xy). It follows that h(y) − h(x) = rxyθ (xy), hence θ (xy) = cxy(h(y) − h(x)), meaning that θ is indeed the current flow associated with h.

Since θ (xy) = cxy(h(y) − h(x)) for any x ∼ y, the node law of immediately implies that h is a voltage. To show that h is unique up to an additive constant, suppose that <sup>g</sup> : <sup>V</sup> <sup>→</sup> <sup>R</sup> is another voltage such that rxyθ (xy) <sup>=</sup> g(y) <sup>−</sup> g(x). It follows that g(y) − h(y) = g(x) − h(x) for any x ∼ y. Since G is connected it follows that g − h is the constant function on V . 

**Definition 2.16** The **strength** of a flow θ is

$$\|\theta\| = \sum\_{\boldsymbol{x} : \boldsymbol{x} \sim \boldsymbol{a}} \theta(a\boldsymbol{x}).$$

*Claim 2.17* For every flow θ,

$$\sum\_{x:x\sim z} \theta(xz) = \|\theta\|.$$

*Proof* We have that

$$\begin{aligned} 0 &= \sum\_{\boldsymbol{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \boldsymbol{\chi}} \theta(\mathbf{x}\mathbf{y}) \\ &= \sum\_{\boldsymbol{x} \in V \backslash \{a, \boldsymbol{z}\}} \sum\_{\mathbf{y} : \mathbf{y} \sim \boldsymbol{\chi}} \theta(\mathbf{x}\mathbf{y}) + \sum\_{\mathbf{y} : \mathbf{y} \sim \boldsymbol{a}} \theta(\mathbf{a}\mathbf{y}) + \sum\_{\mathbf{y} : \mathbf{y} \sim \boldsymbol{\chi}} \theta(\mathbf{z}\mathbf{y}) \\ &= \sum\_{\mathbf{y} : \mathbf{y} \sim \boldsymbol{a}} \theta(\mathbf{a}\mathbf{y}) + \sum\_{\mathbf{y} : \mathbf{y} \sim \boldsymbol{\chi}} \theta(\mathbf{z}\mathbf{y}) \end{aligned}$$

where the first equality is due to antisymmetry, and the third equality is due to the node law. The claim follows again by antisymmetry. 

*Claim 2.18* If θ1, θ<sup>2</sup> are flows satisfying the cycle law and θ1=θ2, then θ<sup>1</sup> = θ2.

*Proof* Let θ¯ = θ<sup>1</sup> − θ2. According to Claim 2.11, θ¯ is a flow. It also satisfies the cycle law, as for every cycle e1,..., em,

$$\sum\_{i=1}^{m} r\_{\boldsymbol{e}\_{i}} \bar{\theta}(\vec{\boldsymbol{e}}\_{i}) = \sum\_{i=1}^{m} r\_{\boldsymbol{e}\_{i}} \theta\_{1}(\vec{\boldsymbol{e}}\_{i}) - \sum\_{i=1}^{m} r\_{\boldsymbol{e}\_{i}} \theta\_{2}(\vec{\boldsymbol{e}}\_{i}) = 0.1$$

Observe in addition that θ¯=θ1−θ2 = 0. Now, let h = hθ¯ be the voltage defined in Claim 2.15, chosen so that h(a) = 0. Note that it is harmonic at a, since

$$\begin{split} \frac{1}{\pi\_{a}} \sum\_{\boldsymbol{x}:\boldsymbol{x}\sim a} c\_{a\boldsymbol{x}} h(\boldsymbol{x}) &= \frac{1}{\pi\_{a}} \sum\_{\boldsymbol{x}:\boldsymbol{x}\sim a} c\_{a\boldsymbol{x}} (h(a) + r\_{a\boldsymbol{x}} \bar{\theta}(a\boldsymbol{x})) \\ &= \frac{1}{\pi\_{a}} \sum\_{\boldsymbol{x}:\boldsymbol{x}\sim a} c\_{a\boldsymbol{x}} h(a) + \frac{1}{\pi\_{a}} \sum\_{\boldsymbol{x}:\boldsymbol{x}\sim a} \bar{\theta}(a\boldsymbol{x}) = h(a) + \frac{\|\bar{\theta}\|}{\pi\_{a}} = h(a). \end{split}$$

Similarly, using Claim 2.17 it is also harmonic at z. Since h is harmonic everywhere, it is constant by Claim 2.5, and thus h ≡ 0, hence θ¯ ≡ 0 and so θ<sup>1</sup> = θ2. 

This last claim prompts the following useful definition.

**Definition 2.19** The **unit current flow** from a to z is the unique current flow from a to z of strength 1.

#### **2.3 The Effective Resistance of a Network**

Suppose we are given a voltage h on a network G with fixed vertices a and z. Scaling h by a constant multiple causes the associated current flow to scale by the same multiple, while adding a constant to h does not change the current flow at all. Therefore, the strength of the current flow is proportional to the difference h(z) − h(a).

*Claim 2.20* For every non-constant voltage h and a current flow θ corresponding to h, the ratio

$$\frac{h(z) - h(a)}{\|\theta\|}\tag{2.4}$$

is a positive constant which does not depend on h.

*Proof* Let h1, h<sup>2</sup> be two non-constant voltages, and let θ1, θ<sup>2</sup> be their associated current flows. For i = 1, 2, let h¯<sup>i</sup> = hi/θi and let θ¯ i be the current flow associated with h¯<sup>i</sup> (note that since hi is non-constant θi = 0). Thus, θ¯ <sup>i</sup> = 1. By Claim 2.18 we get θ¯ <sup>1</sup> = θ¯ <sup>2</sup> and therefore h¯<sup>1</sup> = h¯<sup>2</sup> + c for some constant c by Claim 2.15. It follows that h¯1(z) − h¯1(a) = h¯2(z) − h¯2(a).

To see that this constant is positive, it is enough to check one particular choice of a voltage. By Claim 2.8, let h be the voltage with h(a) = 0 and h(z) = 1. By Claim 2.9 and since G is connected, we have that h(x) > 0 for at least one neighbor x of a. Thus, the corresponding current flow θ has θ > 0 making (2.4) positive. 

**Fig. 2.2** Examples for effective resistances of two networks with unit edge conductances. (**a**) For the voltage depicted, the voltage difference between a and z is 5, and the current flow's strength is 1, hence the effective resistance is 5/1 = 5. (**b**) For the voltage depicted, the voltage difference between a and z is 1, and the current flow's strength is 1, hence the effective resistance is 1/1 = 1

Claim 2.20 is the mathematical manifestation of Ohm's law which states that the voltage difference across an electric circuit is proportional to the current through it. The constant of proportionality is usually called the *effective resistance* of the circuit.

**Definition 2.21** The number defined in (2.4) is called the **effective resistance** between a and z in the network, and is denoted *R*eff(a ↔ z). We call its reciprocal the **effective conductance** between a and z and is denoted *C*eff(a ↔ z) := *<sup>R</sup>*eff(a <sup>↔</sup> z)−1.

For examples of computing the effective resistances of networks, see Fig. 2.2.

**Notation** In most cases we write *R*eff(a ↔ z) and suppress the notation of which network we are working on. However, when it is important to us what the network is, we will write *R*eff(a ↔ z; G) for the effective resistance in the network G with unit edge conductances and *R*eff(a ↔ z;(G,{re})) for the effective resistance in the network G with edge resistances {re}e∈E. Furthermore, given disjoint subsets A and Z of vertices in a graph G, we write *R*eff(A ↔ Z) for the effective resistance between a and z in the network obtained from the original network by identifying all the vertices of A into a single vertex a, and all the vertices of Z into a single vertex z.

**Probabilistic Interpretation** For a vertex x we write τ + <sup>x</sup> for the stopping time

$$
\pi\_\chi^+ = \min \{ t \ge 1 \mid X\_I = x \} \, , \tag{2.5}
$$

where Xt is the weighted random walk on the network, as defined in (2.2). Note that if X<sup>0</sup> = x then τx = τ <sup>+</sup> <sup>x</sup> with probability 1.

*Claim 2.22*

$$\mathcal{R}\_{\text{eff}}(a \leftrightarrow z) = \frac{1}{\pi\_a \mathbb{P}\_a(\mathfrak{r}\_\sharp < \mathfrak{r}\_a^+)} \dots$$

*Proof* Consider the voltage h satisfying h(a) = 0 and h(z) = 1, and let θ be the current flow associated with h. Due to uniqueness of h (Corollary 2.7) we have that

#### 2.3 The Effective Resistance of a Network 19

for x = a,z,

$$h(\mathfrak{x}) = \mathbb{P}\_{\mathfrak{x}}(\mathfrak{r}\_{\mathfrak{z}} < \mathfrak{r}\_a),$$

hence

$$\mathbb{P}\_{a}(\tau\_{\varepsilon} < \tau\_{a}^{+}) = \frac{1}{\pi\_{a}} \sum\_{x \sim a} c\_{ax} \mathbb{P}\_{x}(\tau\_{\varepsilon} < \tau\_{a})$$

$$= \frac{1}{\pi\_{a}} \sum\_{x \sim a} c\_{ax} h(x)$$

$$= \frac{1}{\pi\_{a}} \sum\_{x \sim a} \theta(ax) = \frac{\|\theta\|}{\pi\_{a}} = \frac{1}{\pi\_{a} \mathcal{R}\_{\text{eff}}(a \leftrightarrow z)}. \qquad \square$$

**Network Simplifications** Sometimes a network can be replaced by a simpler network, without changing the effective resistance between a pair of vertices.

*Claim 2.23 (Parallel Law)* Conductances add in parallel. Suppose e1, e<sup>2</sup> are parallel edges between a pair of vertices, with conductances c<sup>1</sup> and c2, respectively. If we replace them with a single edge e with conductance c<sup>1</sup> +c2, then the effective resistance between a and z is unchanged.

A demonstration of the parallel law appears in Fig. 2.3.

*Proof* Let G be the graph where e<sup>1</sup> and e<sup>2</sup> are replaced with e with conductance c<sup>1</sup> + c2. Then it is immediate that if h is any voltage function on G, then it remains a voltage function on the network G . The claim follows. 

*Claim 2.24 (Series Law)* Resistances add in series. Suppose that u ∈ {a,z} is a vertex of degree 2 and that e<sup>1</sup> = (u, v1) and e<sup>2</sup> = (u, v2) are the two edges touching u with edge resistances r<sup>1</sup> and r2, respectively. If we erase u and replace e<sup>1</sup> and e<sup>2</sup> by a single edge e = (v1, v2) of resistance r<sup>1</sup> + r2, then the effective resistance between a and z is unchanged.

The series law is depicted in Fig. 2.4.

*Proof* Denote by G the graph in which u is erased and e<sup>1</sup> and e<sup>2</sup> are replaced by a single edge (v1, v2) of resistance r<sup>1</sup> + r2. Let θ be a current flow from a to z in G, and define a flow θ from a to z in G by putting θ (e) = θ (e) for any e = e1, e<sup>2</sup> and θ (v1, v2) = θ (v1, u). Since u had degree 2, it must be that θ (v1, u) = θ (u, v2).

$$\underbrace{\cdots}\_{u} \asymp\_{c\_{2}} \cdots \asymp\_{c\_{1}} \cdots \asymp\_{c\_{1}} \cdots$$

**Fig. 2.3** Demonstrating the parallel law. Two parallel edges are replaced by a single edge

v1 <sup>u</sup> <sup>v</sup><sup>2</sup> <sup>r</sup><sup>1</sup> <sup>r</sup><sup>2</sup> <sup>v</sup><sup>1</sup> <sup>v</sup><sup>2</sup> r<sup>1</sup> + r<sup>2</sup>

**Fig. 2.4** An example of a network G where edges in series are replaced by a single edge

Thus θ satisfies the node law at any x ∈ {a,z} and θ=θ . Furthermore, since θ satisfies the cycle law, so does θ . We conclude θ is a current flow of the same strength as θ and the voltage difference they induce is the same. 

The operation of **gluing** a subset of vertices S ⊂ V consists of identifying the vertices of S into a single vertex and keeping all the edges and their conductances. In this process we may generate parallel edges or loops.

*Claim 2.25 (Gluing)* Gluing vertices of the same voltage does not change the effective resistance between a and z.

*Proof* This is immediate since the voltage on the glued graph is still harmonic. 

**Example: Spherically Symmetric Tree** Let be a **spherically symmetric tree**, that is, a rooted tree where all vertices at the same distance from the root have the same number of children. Denote by ρ the root of the tree, and let {dn}n∈<sup>N</sup> be a sequence of positive integers. Every vertex at distance n from the root ρ has dn children. Denote by n the set of all vertices of height n. We would like to calculate *R*eff(ρ ↔ n). Due to the tree's symmetry, all vertices at the same level have the same voltage and therefore by Claim 2.25 we can identify them. Our simplified network has now one vertex for each level, denoted by {vi}i∈<sup>N</sup> (where ρ = v0), with |n+1| edges between vn and vn+1. Using the parallel law (Claim 2.23), we can reduce each set of <sup>|</sup>n<sup>|</sup> edges to a single edge with resistance <sup>1</sup> |n| , then, using the series law (Claim 2.24) we get

$$\mathcal{R}\_{\mathrm{eff}}(\rho \leftrightarrow \Gamma\_n) = \sum\_{i=1}^n \frac{1}{|\Gamma\_i|} = \sum\_{i=1}^n \frac{1}{d\_0 \cdots d\_{i-1}},$$

see Fig. 2.5.

By Claim 2.22 we learn that

$$\mathbb{P}\_{\rho}(\mathfrak{r}\_{n} < \mathfrak{r}\_{\rho}^{+}) = \frac{1}{d\_{0} \sum\_{l=1}^{n} \frac{1}{d\_{0} \cdots d\_{l-1}}} \,, \tag{2.6}$$

where τn is the hitting time of n for the random walk on . Observe that

$$\mathbb{P}\_{\rho} \left( \mathfrak{r}\_{\mathbb{M}} < \mathfrak{r}\_{\rho}^{+} \text{ for all } n \right) = \mathbb{P}\_{\rho} \left( X\_{I} \text{ never returns to } \rho \right) \,, .$$

so by (2.6) we reach an interesting dichotomy. If ∞ i=1 1 <sup>d</sup>1···di = ∞, then the random walker returns to ρ with probability 1, and hence returns to ρ infinitely often almost surely. If ∞ i=1 1 <sup>d</sup>1···di <sup>&</sup>lt; <sup>∞</sup>, then with positive probability the walker never returns to ρ, and hence visits ρ only finitely many times almost surely.

The former graph is called a **recurrent** graph and the latter is called **transient**. We will get back to this dichotomy in Sect. 2.5.

#### *The Commute Time Identity*

The following lemma shows that the effective resistance between a and z is proportional to the expected time it takes the random walk starting at a to visit z and then return to a, in other words, the expected *commute time* between a and z. We will use this lemma only in Chap. 6 so the impatient reader may skip this section and return to it later.

**Lemma 2.26 (Commute Time Identity)** *Let* G = (V , E) *be a finite network and* a = z *two vertices. Then*

$$\mathbb{E}\_a[\mathfrak{r}\_\varepsilon] + \mathbb{E}\_\varepsilon[\mathfrak{r}\_a] = 2\mathcal{R}\_{\mathrm{eff}}(a \leftrightarrow z) \sum\_{e \in E} c\_e.$$

*Proof* We denote by Gz : <sup>V</sup> <sup>×</sup> <sup>V</sup> <sup>→</sup> <sup>R</sup> the so-called Green function

Gz(a, x) <sup>=</sup> <sup>E</sup>a[number of visits to <sup>x</sup> before <sup>z</sup>]

and note that

$$\mathbb{E}\_a[\tau\_\varepsilon] = \sum\_{\chi \in V} G\_\varepsilon(a, \chi).$$

It is straightforward to show that the function ν(x) = Gz(a, x)/πx is harmonic in <sup>V</sup> \ {a,z}. Also, we have that Gz(a, z) <sup>=</sup> 0 and Gz(a, a) <sup>=</sup> <sup>1</sup> Pa(τz<τa) = πa*R*eff(a ↔ z). Thus, ν is a voltage function with boundary conditions ν(z) = 0 and ν(a) = *R*eff(a ↔ z) which satisfies

$$\mathbb{E}\_a[\pi\_\varepsilon] = \sum\_{\chi \in V} \nu(\mathfrak{x}) \pi\_\chi \, .$$

Similarly, the same analysis for <sup>E</sup>z[τa] yields the same result, with the voltage function η which has boundary conditions η(z) = *R*eff(a ↔ z) and η(a) = 0. Therefore, η(x) = ν(a)−ν(x) for all x ∈ V since both sides are harmonic functions in V \ {a,z} that receive the same boundary values. This implies that

$$\mathbb{E}\_{\mathbb{Z}}[\mathfrak{r}\_a] = \sum\_{\mathfrak{x} \in V} \pi\_{\mathfrak{x}} \left( \nu(a) - \nu(\mathfrak{x}) \right).$$

Summing these up gives

$$\mathbb{E}\_a[\mathfrak{r}\_\varepsilon] + \mathbb{E}\_\mathfrak{r}[\mathfrak{r}\_a] = \sum\_{\chi \in V} \pi\_\chi \nu(a) = 2 \sum\_{e \in E} c\_e \mathcal{R}\_{\text{eff}}(a \leftrightarrow z). \tag{7}$$

#### **2.4 Energy**

So far we have seen how to compute the effective resistance of a network via harmonic functions and current flows. However, in typical situations it is hard to find a flow satisfying the circle law. Luckily, an extremely useful property of the effective resistance is that it can be represented by a variational problem. Our intuition from highschool physics suggests that the *energy* of the unit current flow is minimal among all unit flows from a to z. The notion of energy can be made precise and will allow us to obtain valuable monotonicity properties. For instance, removing any edge from an electric network can only increase its effective resistance. Hence, any recurrent graph remains recurrent after removing any subset of edges from it. Two variational problems govern the effective resistance, Thomson's principle, which is used to bound the effective resistance from above, and Dirichlet's principle, used to bound it from below.

**Definition 2.27** The **energy** of a flow θ from a to z, denoted by *E*(θ ), is defined to be

$$\mathcal{E}(\theta) := \frac{1}{2} \sum\_{\vec{e} \in \vec{E}} r\_{\vec{e}} \,\theta(\vec{e})^2 = \sum\_{e \in E} \theta(e)^2 r\_e.$$

Note that in the second sum we sum over undirected edges, but since θ (xy)<sup>2</sup> <sup>=</sup> θ (yx)2, this is well defined.

#### **Theorem 2.28 (Thomson's Principle)**

$$\mathcal{R}\_{\text{eff}}(a \leftrightarrow z) = \inf \{ \mathcal{E}(\theta) : \|\theta\| = 1, \theta \text{ is a flow from } a \text{ to } z \} $$

*and the unique minimizer is the unit current flow.*

*Proof* First, we will show that the energy of the unit current flow is the effective resistance. Let I be the unit current flow, and h the corresponding (Claim 2.15) voltage function.

$$\begin{split} \mathcal{E}(I) &= \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} \colon \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} I(\mathbf{x} \mathbf{y})^2 = \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} \colon \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} \left( \frac{h(\mathbf{y}) - h(\mathbf{x})}{r\_{\mathbf{xy}}} \right) I(\mathbf{x} \mathbf{y}) \\ &= \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} \colon \mathbf{y} \sim \mathbf{x}} \left( h(\mathbf{y}) - h(\mathbf{x}) \right) I(\mathbf{x} \mathbf{y}) \\ &= \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} \colon \mathbf{y} \sim \mathbf{x}} h(\mathbf{y}) I(\mathbf{x} \mathbf{y}) - \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} \colon \mathbf{y} \sim \mathbf{x}} h(\mathbf{x}) I(\mathbf{x} \mathbf{y}). \end{split}$$

Observe that in the second term of the right hand side, for every x = a,z the sum over all <sup>y</sup> <sup>∼</sup> <sup>x</sup> is 0 due to the node law, hence the entire term equals <sup>1</sup> <sup>2</sup> (h(a)−h(z)). From antisymmetry of <sup>I</sup> , the first term on the right hand side equals <sup>−</sup><sup>1</sup> <sup>2</sup> (h(a) − h(z)), hence the right hand side equals altogether h(z) − h(a) = *R*eff(a ↔ z).

We will now show that every other flow J with J = 1 has *E*(J ) ≥ *E*(I ). Let J be such flow and write J = I + (J − I ). Set θ = J − I and note that θ = 0. We have

$$\begin{split} \mathcal{E}(J) &= \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} (I(\mathbf{x} \mathbf{y}) + \theta(\mathbf{x} \mathbf{y}))^2 \\ &= \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} I(\mathbf{x} \mathbf{y})^2 + \frac{1}{2} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} \theta(\mathbf{x} \mathbf{y})^2 \\ &+ \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} \theta(\mathbf{x} \mathbf{y}) I(\mathbf{x} \mathbf{y}) \\ &= \mathcal{E}(I) + \mathcal{E}(\theta) + \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} \theta(\mathbf{x} \mathbf{y}) I(\mathbf{x} \mathbf{y}). \end{split}$$

Now,

$$\begin{aligned} \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} \theta(\mathbf{xy}) I(\mathbf{xy}) &= \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} r\_{\mathbf{xy}} \theta(\mathbf{xy}) \frac{(h(\mathbf{y}) - h(\mathbf{x}))}{r\_{\mathbf{xy}}} \\ &= \sum\_{\mathbf{x} \in V} \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} \theta(\mathbf{xy}) (h(\mathbf{y}) - h(\mathbf{x})) \\ &= 2 \cdot \|\theta\| \cdot (h(\mathbf{z}) - h(\mathbf{a})) = 0, \end{aligned}$$

where the last inequality follows from the same reasoning as before. We conclude that *E*(J ) ≥ *E*(I ) as required and that equality holds if and only if *E*(θ ) = 0, that is, if and only if J = I . 

**Corollary 2.29 (Rayleigh's Monotonicity Law)** *If* {re}e∈<sup>E</sup> *and* {r <sup>e</sup>}e∈<sup>E</sup> *are edge resistances on the same graph* G *so that* re ≤ r <sup>e</sup> *for all edges* e ∈ E*, then*

$$
\mathcal{R}\_{\text{eff}}(a \leftrightarrow z; (G, \{r\_{\varepsilon}\})) \leq \mathcal{R}\_{\text{eff}}(a \leftrightarrow z; (G, \{r'\_{\varepsilon}\})).
$$

*Proof* Let θ be a flow on G, then

$$\sum\_{e \in E} r\_e \theta(e)^2 \le \sum\_{e \in E} r'\_e \theta(e)^2.$$

This inequality is preserved while taking infimum over all flows with strength 1. Applying Theorem 2.28 finishes the proof. 

**Corollary 2.30** *Gluing vertices cannot increase the effective resistance between* a *and* z*.*

*Proof* Denote by G the original network and by G the network obtained from gluing a subset of vertices. Then every flow θ on G (viewed as a function on the edges) is a flow on G . Hence the infimum in Theorem 2.28 taken over flows in G is taken over a larger subset of flows. 

**Definition 2.31** The **energy** of a *function* <sup>h</sup> : <sup>V</sup> <sup>→</sup> <sup>R</sup>, denoted by *<sup>E</sup>*(h), is defined to be

$$\mathcal{E}(h) := \sum\_{\{\mathbf{x}, \mathbf{y}\} \in E} c\_{\mathbf{x}\mathbf{y}} (h(\mathbf{x}) - h(\mathbf{y}))^2.$$

Compare the following lemma with Thomson's principle (Theorem 2.28).

**Lemma 2.32 (Dirichlet's Principle)** *Let* G *be a finite network with source* a *and sink* z*. Then*

$$\frac{1}{\mathcal{R}\_{\rm eff}(a \leftrightarrow z)} = \inf \left\{ \mathcal{E}(h) : h : V \to \mathbb{R}, \ h(a) = 0, \ h(z) = 1 \right\}.$$

*Proof* The infimum is obtained when h is *the* harmonic function taking 0 and 1 at a,z respectively. The reason is that if there exists v = a,z with

$$h(v) \neq \sum\_{u \sim v} \frac{c\_{vu}}{\pi\_v} h(u),\tag{2.7}$$

then we can change the value of h at v to be the right hand side of (2.7) and the energy will only decrease. One way to see this is that if X is a random variable with a second moment, then the value <sup>E</sup>(X) minimizes the function f (x) <sup>=</sup> <sup>E</sup> (X <sup>−</sup> x)<sup>2</sup> .

Let h be that harmonic function and let I be its current flow, so I (xy) = cxy(h(y) − h(x)). Write Iˆ = *R*eff(a ↔ z)· I , so Iˆ = 1. By Thomson's principle,

$$\mathcal{R}\_{\text{eff}}(a \leftrightarrow z) = \mathcal{E}(\hat{I}) = \sum\_{e \in E} r\_e \hat{I}(e)^2 = \sum\_{\{\mathbf{x}, \mathbf{y}\} \in E} r\_{\mathbf{x}\mathbf{y}} \mathcal{R}\_{\text{eff}}(a \leftrightarrow z)^2 c\_{\mathbf{x}\mathbf{y}}^2 (h(\mathbf{y}) - h(\mathbf{x}))^2,$$

hence

$$\frac{1}{\mathcal{R}\_{\rm eff}(a \leftrightarrow z)} = \mathcal{E}(h). \tag{7}$$

#### **2.5 Infinite Graphs**

Let G = (V , E) be an infinite connected graph with edge resistances {re}e∈E. We assume henceforth that this network is **locally finite**, that is, for any vertex x ∈ V we have <sup>y</sup>:y∼<sup>x</sup> cxy <sup>&</sup>lt; <sup>∞</sup>. Let {Gn} be a sequence of finite subgraphs of <sup>G</sup> such that <sup>n</sup>∈<sup>N</sup> Gn <sup>=</sup> <sup>G</sup> and Gn <sup>⊂</sup> Gn+1; we call such a sequence an **exhaustive sequence** of G. Identify all vertices of G \ Gn with a single vertex zn.

*Claim 2.33* Given an exhaustive sequence {Gn} of G, the limit

$$\lim\_{n \to \infty} \mathcal{R}\_{\text{eff}}(a \leftrightarrow z\_n; G\_n \cup \{z\_n\}) \tag{2.8}$$

exists.

*Proof* The graph Gn ∪ {zn} can be obtained from Gn+<sup>1</sup> ∪ {zn+1} by gluing the vertices in Gn+<sup>1</sup> \ Gn with zn+<sup>1</sup> and labeling the new vertex zn. By Corollary 2.30, the effective resistance *R*eff(a ↔ zn; Gn ∪ {zn}) is increasing in n. 

*Claim 2.34* The limit in (2.8) does not depend on the choice of exhaustive sequence {Gn}.

*Proof* Indeed, let {Gn} and {G <sup>n</sup>} be two exhaustive sequences of G. We can find subsequences {ik}k≥<sup>1</sup> and {jk}k≥<sup>1</sup> such that

$$G\_{\ell\_1} \subseteq G\_{\jmath\_1}' \subseteq G\_{\ell\_2} \subseteq \dotsb$$

Since {Gi<sup>1</sup> , G j1 , Gi<sup>2</sup> ,...} is itself an exhaustive sequence of G, the limit of effective resistances for this sequence exists and equals the limits of effective resistances for the subsequences {Gik } and {G jk }. In turn, these are equal to the limits of effective resistances for the original sequences {Gn} and {G <sup>n</sup>}, respectively. 

**Definition 2.35** In an infinite network, the effective resistance from a vertex a and ∞ is

$$\mathcal{R}\_{\mathrm{eff}}(a \leftrightarrow \infty) := \lim\_{n \to \infty} \mathcal{R}\_{\mathrm{eff}}(a \leftrightarrow z\_n; G\_n \cup \{z\_n\})\,.$$

We are now able to address the question of recurrence versus transience of a graph systematically. Recall the definition of τ + <sup>x</sup> in (2.5). In an infinite network we define τ + <sup>a</sup> = ∞ when there is no time t ≥ 1 such that Xt = a.

**Definition 2.36** A network (G,{re}e∈E) is called **recurrent** if <sup>P</sup>a(τ <sup>+</sup> <sup>a</sup> = ∞) = 0, that is, if the probability of the random walker started at a never returning to a is 0. Otherwise, it is called **transient** .

Observe that since G is connected, if Pa(τ <sup>+</sup> <sup>a</sup> = ∞) = 0 for one vertex a, then it holds for all vertices in the network. As we have seen, if n is large enough so that a ∈ Gn, then

$$\mathcal{R}\_{\text{eff}}(a \leftrightarrow z\_n; G\_n \cup \{z\_n\}) = \frac{1}{\pi\_a \cdot \mathbb{P}\_a\left(\mathfrak{r}\_{G \backslash G\_n} < \mathfrak{r}\_a^+\right)}\ .$$

Since <sup>n</sup>{τG\Gn < τ <sup>+</sup> <sup>a</sup> }={τ <sup>+</sup> <sup>a</sup> = ∞} we have

$$\mathcal{R}\_{\rm eff}(a \leftrightarrow \infty) = \frac{1}{\pi\_a \cdot \mathbb{P}\_a(\mathfrak{r}\_a^+ = \infty)}\,,$$

with the convention that 1/0 = ∞.

**Definition 2.37** Let <sup>G</sup> be an infinite network. A function <sup>θ</sup> : E(G) <sup>→</sup> <sup>R</sup> is a **flow from** a **to** ∞ if it is anti-symmetric and satisfies the node law on each vertex v = a.

The following follows easily from Theorem 2.28, we omit the proof.

**Theorem 2.38 (Thomson's Principle for Infinite Networks)** *Let* G *be an infinite network, then*

*R*eff(a ↔ ∞) = inf{*E*(θ ) : θ *is a flow from* a *to* ∞ *of strength* 1}.

**Corollary 2.39** *Let* G *be an infinite graph. The following are equivalent:*


We will now develop a useful method for bounding effective resistances from below. This will lead us to a popular sufficient criterion for recurrence in Corollary 2.43.

**Definition 2.40** A **cutset** ⊆ E(G) separating a from z is a set of edges such that every path from a to z must use an edge from .

*Claim 2.41* Let θ be a flow from a to z in a finite network, and let a cutset separating a from z. Then

$$\sum\_{e \in \Gamma} |\theta(e)| \ge \|\theta\|.$$

*Proof* Denote by Z the set of vertices separated from a by . Denote by G the network where Z is identified to a single vertex x and all edges having both endpoints in Z are removed. Now, the restriction of θ to the edges of the new network is a flow from a to x. By Claim 2.17, we have <sup>y</sup>:y∼<sup>x</sup> θ (yx) = θ. Also, all edges incident to x must be in , since otherwise x is not separated from a by . Therefore

$$\sum\_{e \in \Gamma} |\theta(e)| \ge \sum\_{\mathbf{y} : \mathbf{y} \sim \mathbf{x}} \theta(\mathbf{y} \mathbf{x}) = \|\theta\|. \tag{7}$$

**Theorem 2.42 (Nash-Williams Inequality)** *Let* {n} *be disjoint cutsets separating* a *from* z *in a finite network. Then*

$$\mathcal{R}\_{\text{eff}}(a \leftrightarrow z) \ge \sum\_{n} \left( \sum\_{e \in \Gamma\_n} c\_e \right)^{-1} \cdot \mathbf{1}$$

*Proof* Let θ be a flow from a to z with θ = 1. From Cauchy-Schwarz, for each n we have

$$\left(\sum\_{e\in\Gamma\_n} \sqrt{r\_e}\sqrt{c\_e}|\theta(e)|\right)^2 \le \sum\_{e\in\Gamma\_n} c\_e \sum\_{e\in\Gamma\_n} r\_e \theta(e)^2 \dots$$

Also, since n is a cutset, the flow passing through n is at least θ, by Claim 2.41. So

$$\left(\sum\_{e \in \Gamma\_n} \sqrt{r\_e} \sqrt{c\_e} |\theta(e)|\right)^2 \ge \|\theta\|^2 = 1.$$

Combining them, we get that

$$\sum\_{e \in \Gamma\_n} r\_e \theta(e)^2 \ge \frac{1}{\sum\_{e \in \Gamma\_n} c\_e}.$$

Summing over all n gives

$$\mathcal{E}(\theta) \ge \sum\_{n} \sum\_{e \in \Gamma\_n} r\_e \theta(e)^2 \ge \sum\_{n} \left(\sum\_{e \in \Gamma\_n} c\_e\right)^{-1}.$$

Applying Thomson's principle (Theorem 2.28) yields the result. 

Consider now an infinite network G = (V , E). We say that ⊂ E is a cutset separating a from ∞ if any infinite simple path from a must intersect .

**Corollary 2.43** *In any infinite network, if there exists a collection* {n} *of disjoint cutsets separating* a *from* ∞ *such that*

$$\sum\_{n} \left( \sum\_{e \in \Gamma\_n} c\_e \right)^{-1} = \infty,$$

*then the network is recurrent.*

*Example 2.44 (*Z<sup>2</sup> *is Recurrent)* Define n as the set of vertical edges {(x, y), (x, y +1)} with |x| ≤ n and min{|y|,|y +1|} = n along with the horizontal edges {(x, y), (x + 1,y)} with |y| ≤ n and min{|x|,|x + 1|} = n, see Fig. 2.6. Then

**Fig. 2.6** A part of Z2: the edges in {−1, 0, 1} <sup>2</sup> are drawn in bold. <sup>1</sup> is dashed

{n} is a collection of disjoint cutsets separating 0 from ∞. Also, |n| = 4(2n + 1) and therefore n <sup>e</sup>∈<sup>n</sup> ce <sup>−</sup><sup>1</sup> = ∞. We deduce by Corollary 2.43 that <sup>Z</sup><sup>2</sup> is recurrent.

*Remark 2.45* There are recurrent graphs for which there exists M < ∞ such that for every collection {n} of disjoint cutsets, n <sup>e</sup>∈<sup>n</sup> ce <sup>−</sup><sup>1</sup> ≤ M. Therefore, the Nash-Williams inequality is not sharp. See Exercise 4 of this chapter.

#### **2.6 Random Paths**

We now present the method of random paths, which is one of the most useful methods for generating unit flows on a network and bounding their energy. In fact, it is possible to show that the electric flow can be represented by such a random path. Suppose G is a network with fixed vertices a,z and μ is a probability measure on the set of paths from a to z.

*Claim 2.46* For a path γ sampled from μ, let

θγ (e) = (# of times e was traversed by γ ) − (# of times ewas traversed by γ ),

where by e and ewe mean the two orientations of an edge e of G. Set

$$
\theta\left(\vec{e}\right) = \mathbb{E}\theta\_\mathcal{Y}\left(\vec{e}\right).
$$

Then θ is a flow from a to z with θ = 1.

*Proof* θ is antisymmetric since θγ is antisymmetric for every γ , and it satisfies the node law since θγ satisfies the node law. Similarly, the "strength" of θγ (i.e., <sup>x</sup>∼<sup>a</sup> θγ (ax)) is 1, hence θ = 1. 

An example of the use of this method is the following classical result.

**Theorem 2.47** Z<sup>3</sup> *is transient.*

*Proof* For R > 0 denote by BR = {(x, y, z) : <sup>x</sup><sup>2</sup> <sup>+</sup> <sup>y</sup><sup>2</sup> <sup>+</sup> <sup>z</sup><sup>2</sup> <sup>≤</sup> <sup>R</sup>2} the ball of radius <sup>R</sup> in <sup>R</sup>3. Put VR <sup>=</sup> BR <sup>∩</sup>Z<sup>3</sup> and let ∂VR be the external vertex boundary of VR, that is, the set of vertices not in VR which belong to an edge with an endpoint in VR.

We construct a random path μ from the origin **0** to ∂VR by choosing a uniform random point **<sup>p</sup>** in ∂BR = {(x, y, z) : <sup>x</sup><sup>2</sup> <sup>+</sup> <sup>y</sup><sup>2</sup> <sup>+</sup> <sup>z</sup><sup>2</sup> <sup>=</sup> <sup>R</sup>2}, drawing a straight line between **0** and **p** in R3, considering the set of distance at most 10 in R<sup>3</sup> from the line, and then choosing (in some arbitrary fashion) a path in Z<sup>3</sup> which is contained inside this set. The non-optimal constant 10 was chosen in order to guarantee that such a discrete path exists for any point **p** ∈ ∂BR.

By Claim 2.46, the measure μ corresponds to a flow from **0** to ∂VR. To estimate the energy of this flow, we note that if e is an edge at distance r ≤ R from the origin, then the probability that it is traversed by a path drawn by μ is O(r−2). Furthermore, there are O(r2) such edges. Hence the energy of the flow is at most

$$\mathcal{E} = O\left(\sum\_{r=1}^{R} r^2 \cdot (r^{-2})^2\right) \le \mathcal{C} \llcorner$$

for some constant C < ∞ which does not depend on R. By Claim 2.46 and Theorem 2.28 we learn that *R*eff(**0** ↔ ∂VR) ≤ C for all R, and so by Corollary 2.39 we deduce that <sup>Z</sup><sup>3</sup> is transient. 

#### **2.7 Exercises**

1. Let Gz(a, x) be the Green's function, that is,

$$G\_z(a, x) = \mathbb{E}\_a \left[ \mathfrak{H} \text{visits to } x \text{ before visiting } z \right].$$

Show that the function h(x) = Gz(a, x)/π(x) is a voltage.

2. Show that the effective resistance satisfies the triangle inequality. That is, for any three vertices x,y,z we have

$$\mathcal{R}\_{\rm eff}(\mathbf{x}\leftrightarrow\mathbf{z}) \le \mathcal{R}\_{\rm eff}(\mathbf{x}\leftrightarrow\mathbf{y}) + \mathcal{R}\_{\rm eff}(\mathbf{y}\leftrightarrow\mathbf{z})\,. \tag{2.9}$$

3. Let a,z be two vertices of a finite network and let τa, τz be the first visit time to a and z, respectively, of the weighted random walk. Show that for any vertex x

$$\mathbf{P}\_{\mathbf{x}}(\mathfrak{r}\_{a} < \mathfrak{r}\_{\mathfrak{z}}) \leq \frac{\mathcal{R}\_{\mathbf{eff}}(\mathfrak{x} \leftrightarrow \{a, z\})}{\mathcal{R}\_{\mathbf{eff}}(\mathfrak{x} \leftrightarrow a)} \,.$$

	- (a) Show that T is recurrent.
	- (b) Show that for any disjoint edge cutsets n we have that <sup>n</sup> |n| <sup>−</sup><sup>1</sup> <sup>&</sup>lt; <sup>∞</sup>. (So, the Nash-Williams criterion for recurrence is not sharp)

resistances are 1. Show that

$$\mathcal{R}\_{\mathrm{eff}}(a \leftrightarrow z; G) = \frac{1}{\mathcal{R}\_{\mathrm{eff}}(a^\* \leftrightarrow z^\*; G^\*)} \cdot \frac{1}{\cdot}$$


$$E\_k = \left\{ \left( 2^k (n - 1/2), 2^k (n + 1/2) \right) : n \in \mathbb{Z} \right\} \dots$$

Is G recurrent or transient?

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# **Chapter 3 The Circle Packing Theorem**

#### **3.1 Planar Graphs, Maps and Embeddings**

**Definition 3.1** A graph G = (V , E) is **planar** if it can be **properly drawn** in the plane, that is, if there exists a mapping sending the vertices to distinct points of R<sup>2</sup> and edges to continuous curves between the corresponding vertices so that no two curves intersect, except at the vertices they share. We call such a mapping a **proper drawing** of G.

*Remark 3.2* A single planar graph has infinitely many drawings. Intuitively, some may seem similar to one another, while others seem different. For example,

The following definition gives a precise sense to the above intuitive equivalence/non-equivalence of drawings.

**Definition 3.3** A **planar map** is a graph endowed with a cyclic permutation of the edges incident to each vertex, such that there exists a proper drawing in which the clockwise order of the curves touching the image of a vertex respects that cyclic permutation.

The combinatorial structure of a planar map allows us to define faces directly (that is, without mentioning the drawing). Consider each edge of the graph as directed in both ways, and say that a directed edge e **precedes** f (or, equivalently,

**Fig. 3.1** Examples for the edge precedence relation. (**a**) e precedes f. (**b**) (x, y) precedes (y, x)

f **succeeds** e), if there exist vertices v, x, y such that e = (x, v), f = (v, y), and y is the successor of x in the cyclic permutation σv; see Fig. 3.1.

We say that e, f **belong to the same face** if there exists a finite directed path e1,..., e<sup>m</sup> in the graph with e<sup>i</sup> preceding ei+<sup>1</sup> for i = 1,...,m − 1 and such that either e = e<sup>1</sup> and f = em, or f = e<sup>1</sup> and e = em. This is readily seen to be an equivalence relation and we call each equivalence class a **face**. Even though a face is a set of directed edges, we frequently ignore the orientations and consider a face as the set of corresponding undirected edges. Each (undirected) edge is henceforth incident to either one or two faces.

When the map is finite an equivalent definition of a face is the set of edges that bound a connected component of the complement of the drawing, that is, of R<sup>2</sup> minus the images of the vertices and edges. This definition is not suitable for infinite planar maps since there may be a complicated set of accumulation points. Given a proper drawing of a finite planar map, there is a unique unbounded connected component of the complement of the drawing; the edges that bound it are called the **outer face** and all other faces are called **inner faces** . However, for any face in a finite map there is a drawing so that this face bounds the unique unbounded connected component, and because of this we shall henceforth refer to the outer face as an arbitrarily chosen face of the map.

We will use the following classical formula.

**Theorem 3.4 (Euler's Formula)** *Suppose* G *is a planar graph with* n *vertices,* m *edges and* f *faces. Then*

$$n - m + f = 2.$$

We now state the main theorem we will discuss and use throughout this course. Its proof is presented in the next section.

**Theorem 3.5 (The Circle Packing Theorem [51])** *Given any finite simple planar map* <sup>G</sup> <sup>=</sup> (V , E)*,* <sup>V</sup> = {v1,...,vn}*, there exist* <sup>n</sup> *circles in* <sup>R</sup>2*,* <sup>C</sup>1,...,Cn*, with disjoint interiors, such that* Ci *is tangent to* Cj *if and only if* {i, j } ∈ E*.*

**Fig. 3.2** Two distinct planar maps (of the same graph) with corresponding circle packings

*Furthermore, for every vertex* vi*, the clockwise order of the circles tangent to* Ci *agrees with the cyclic permutation of* vi*'s neighbors in the map.*

Figure 3.2 gives examples for embeddings of maps which respect the cyclic orderings of neighbors, as guaranteed to exist according to the theorem.

First note that it suffices to prove the theorem for **triangulations**, that is, simple planar maps in which every face has precisely three edges. Indeed, in any planar map we may add a single vertex inside each face and connect it to all vertices bounding that face. The obtained map is a triangulation, and after applying the circle packing theorem for triangulations, we may remove the circles corresponding to the added vertices, obtaining a circle packing of the original map which respects its cyclic permutations. This is depicted in Fig. 3.3.

Thus, it suffices to prove Theorem 3.5 for finite triangulations. In this case an important uniqueness statement also holds.

**Fig. 3.3** Circle packing of a triangulation of a planar map. (**a**) A planar map and a triangulation. (**b**) A circle packing of the triangulation

**Theorem 3.6** *Let* G = (V , E) *be a finite triangulation on vertex set* V = {v1,...,vn} *and assume that* {v1, v2, v3} *form a face. Then for any three positive numbers* ρ1, ρ2, ρ3*, there exists a circle packing* C1,...,Cn *as in Theorem 3.5 with the additional property that* C1, C2, C<sup>3</sup> *are mutually tangent, form the outer face, and have radii* ρ1, ρ2, ρ3*, respectively. Furthermore, this circle packing is unique, up to translations and rotations of the plane.*

#### **3.2 Proof of the Circle Packing Theorem**

We prove Theorem 3.6 which implies Theorem 3.5 as explained above. Therefore we assume from now on that our map is a triangulation. Denote by n, m and f the number of vertices, edges and faces of the map respectively, and observe that 3f = 2m since each edge is counted in exactly two faces, and each face is bounded by exactly three edges. Therefore, by Euler's formula (Theorem 3.4), we have that

$$2 = n - m + f = n - \frac{3}{2}f + f = n - \frac{1}{2}f,$$

thus

$$f = 2n - 4.\tag{3.1}$$

We assume the vertex set is {v1,...,vn}, that {v1, v2, v3} is the outer face and that ρ1, ρ2, ρ<sup>3</sup> are three positive numbers that will be the radii of the outer circles C1, C2, C<sup>3</sup> eventually. Denote by F◦ the set of inner faces of the map, and for a subset of vertices A let F (A) be the set of inner faces with at least one vertex in A. We write F (v) when we mean F ({v}).

Given a vector **<sup>r</sup>** <sup>=</sup> (r1,...,rn) <sup>∈</sup> (0,∞)n, an inner face <sup>f</sup> <sup>∈</sup> <sup>F</sup>◦ bounded by the vertices vi, vj , vk , and a distinguished vertex vj , we associate a number α**r** <sup>f</sup> (vj ) = vivj vk ∈ (0,π) which is the angle of vj in the triangle vivj vk created by connecting the centers of three mutually tangent circles Ci, Cj , Ck of radii ri, rj and rk (that is, in a triangle with side lengths ri + rj , rj + rk and rk + ri). This number can be calculated using the cosine formula

$$
\cos(\angle v\_l v\_j v\_k) = 1 - \frac{2r\_l r\_k}{(r\_l + r\_j)(r\_j + r\_k)},
$$

however, we will not use this formula directly. For every j ∈ {1,...,n} we define

$$\sigma\_{\mathbf{r}}(v\_j) = \sum\_{f \in F(v\_j)} \alpha\_f^{\mathbf{r}}(v\_j).$$

to be the *sum of angles* at vi with respect to **r**. Let θ1, θ2, θ<sup>3</sup> be the angles formed at the centers of three mutually tangent circles C1, C2, C<sup>3</sup> of radii ρ1, ρ2, ρ3. Equivalently, these are the angles of a triangle with edge lengths r<sup>1</sup> + r2, r<sup>2</sup> + r<sup>3</sup> and r<sup>1</sup> + r3. If the vector **r** was the vector of radii of a circle packing of the map satisfying Theorem 3.6, then it would hold that

$$\sigma\_{\mathbf{r}}(v\_{l}) = \begin{cases} \theta\_{l} & i \in \{1, 2, 3\}, \\ 2\pi & \text{otherwise} \end{cases} \tag{3.2}$$

and additionally (r1, r2, r3) = (ρ1, ρ2, ρ3). The proof is split into three parts:


#### *Proof of Theorem 3.6, Step 1: Finding the Radii Vector* **r**

**Observation 3.7** *For every* **r***,*

$$\sum\_{l=1}^{n} \sigma\_{\mathbf{f}}(v\_l) = \left| F^{\diamond} \right| \pi = (2n - 5)\pi.$$

*Proof* Follows immediately since each inner face f bounded by the vertices vi, vj , vk contributes the three angles α**<sup>r</sup>** <sup>f</sup> (vi), <sup>α</sup>**<sup>r</sup>** <sup>f</sup> (vj ) and <sup>α</sup>**<sup>r</sup>** <sup>f</sup> (vk) which sum to π. By (3.1), there are 2n − 5 inner faces. 

We now set

$$\delta\_{\mathbf{r}}(\upsilon\_{i}) = \begin{cases} \sigma\_{\mathbf{r}}(\upsilon\_{i}) - \theta\_{i} & j \in \{1, 2, 3\}, \\ \sigma\_{\mathbf{r}}(\upsilon\_{j}) - 2\pi & \text{otherwise.} \end{cases} \tag{3.3}$$

Using this notation, our goal is to find **r** for which δ**<sup>r</sup>** ≡ 0. It follows from Observation 3.7 that for every **r**,

$$\sum\_{l=1}^{n} \delta\_{\mathbf{r}}(v\_l) = \sum\_{l=1}^{n} \sigma\_{\mathbf{r}}(v\_l) - \theta\_1 - \theta\_2 - \theta\_3 - (n-3) \cdot 2\pi = 0 \,. \tag{3.4}$$

We define

$$\mathcal{E}\_{\mathbf{r}} = \sum\_{l=1}^{n} \delta\_{\mathbf{r}} (v\_l)^2.$$

**Fig. 3.4** When the radius of a circle corresponding to a vertex increases, while the radii of the circles corresponding to its two neighbors in a given face decrease, the vertex's angle in the corresponding triangle decreases (see Observation 3.8)

We would like to find **r** for which *E***<sup>r</sup>** = 0. We will use the following geometric observation; see Fig. 3.4.

**Observation 3.8** *Let* **r** = (r1,...,rn) *and* **r** = (r 1,...,r n)*, and let* f ∈ F◦ *be bounded by* vi, vj , vk *.*


*Proof* A proof using the cosine formula is routine and is omitted. 

We now define an iterative algorithm, whose input and output are both vectors of radii normalized to have <sup>1</sup> norm 1. We start with the vector **<sup>r</sup>**(0) <sup>=</sup> 1 <sup>n</sup> ,..., <sup>1</sup> n , and given **<sup>r</sup>** <sup>=</sup> **<sup>r</sup>**(t ) we construct **<sup>r</sup>** <sup>=</sup> **<sup>r</sup>**(t+1) . Write δ = δ**<sup>r</sup>** and δ = δ**r** , and similarly *E* = *E***<sup>r</sup>** and *E* = *E***r** . We begin by ordering the set of reals {δ(vi) | 1 ≤ i ≤ n}. If <sup>δ</sup> <sup>≡</sup> 0 we are done; otherwise, we may choose <sup>s</sup> <sup>∈</sup> <sup>R</sup> such that the set <sup>S</sup> = {<sup>v</sup> <sup>|</sup> δ(v) > s} = <sup>∅</sup> and its complement <sup>V</sup> \ <sup>S</sup> are non-empty and such that the gap

$$\text{gap}\_{\delta}(\mathcal{S}) := \min\_{\upsilon \in \mathcal{S}} \delta(\upsilon) - \max\_{\upsilon \notin \mathcal{S}} \delta(\upsilon) > 0$$

is maximal over all such s. See Fig. 3.5 for illustration.

**Fig. 3.5** Left: finding the maximum gap between two consecutive values of δ, and splitting the set of values into S and its complement. Right: moving from **r** to **r** closes the gap between S and V \ S

Once we choose S, a step of the algorithm consists of two steps:

1. For some λ ∈ (0, 1) to be chosen later, we set

$$(\mathbf{r}\_{\lambda})\_{l} = \begin{cases} r\_{l} & v\_{l} \in \mathcal{S}, \\ \lambda r\_{l} & v\_{l} \notin \mathcal{S}. \end{cases}$$

2. We normalize **r**<sup>λ</sup> so that the sum of entries is 1, letting **r**¯<sup>λ</sup> be the normalized vector. Note that this step does not change the vector δ.

We will choose an appropriate λ that will decrease all values of δ(v) for v ∈ S, increase all values of δ(v) for v /∈ S, and will close the gap. This is made formal in the following two claims.

*Claim 3.9* For every λ ∈ (0, 1), setting **r** = **r**¯λ, we have that δ (v) ≤ δ(v) for any v ∈ S, and δ (v) ≥ δ(v) for any v /∈ S.

*Claim 3.10* There exists λ ∈ (0, 1) such that setting **r** = **r**¯<sup>λ</sup> gives that gapδ(S) = 0.

*Proof of Claim 3.9* Consider vj ∈/ S and an inner face vi, vj , vk .


It follows that δ (v) ≥ δ(v) for any v /∈ S. An identical argument shows that δ (v) ≤ δ(v) for all v ∈ S. 

In order to prove Claim 3.10, we present another claim.

*Claim 3.11*

$$\lim\_{\lambda \searrow 0} \sum\_{\upsilon \notin \mathcal{S}} \delta\_{\mathbf{r}\_{\lambda}}(\upsilon) > 0.$$

*Proof of Claim 3.10 Using Claim 3.11* The function <sup>λ</sup> → gapδ**r**<sup>λ</sup> (S) is continuous on (0, 1] by the third bullet of Observation 3.8, and its value at λ = 1 is gap<sup>δ</sup> (S) > 0. Claim 3.11 says that if μ > 0 is small enough, then

$$\sum\_{\upsilon \notin \mathcal{S}} \delta\_{\mathbf{r}\_{\mu}}(\upsilon) > 0,$$

from which it follows that maxv /∈<sup>S</sup> δ**r**μ(v) > 0. By (3.4), we also have

$$\sum\_{\upsilon \in S} \delta\_{\mathbf{r}\_{\mu}}(\upsilon) < 0,$$

meaning that minv∈<sup>S</sup> δ**r**μ(v) < 0 and therefore gapδ**r**<sup>μ</sup> (S) < 0. By continuity, there exists <sup>λ</sup> <sup>∈</sup> (μ, <sup>1</sup>) such that gapδ**r**<sup>λ</sup> (S) = 0. 

*Proof of Claim 3.11* We first show that for each face f ∈ F (V \ S) bounded by vi, vj , vk, the sum of angles at the vertices belonging to V \ S converges to π as λ 0. We show this by the following case analysis. The statements in cases II and III can be justified by drawing a picture or appealing to the cosine formula.


It follows that

$$\lim\_{\lambda \searrow 0} \sum\_{\upsilon \notin S} \sigma\_{\mathbf{r}\_{\lambda}}(\upsilon) = |F(V \nmid S)|\pi. \tag{3.5}$$

For convenience, set

$$\theta(v\_i) = \begin{cases} \theta\_i & 1 \le i \le 3, \\ 2\pi & \text{otherwise,} \end{cases}$$

so that δ**r**(v) = σ**r**(v) − θ (v) for all v ∈ V . Then

$$\lim\_{\lambda \searrow 0} \sum\_{\upsilon \notin S} \delta\_{\mathbf{r}\_{\lambda}}(\upsilon) = |F(V \nmid S)| \pi - \sum\_{\upsilon \notin S} \theta(\upsilon). \tag{3.6}$$

Let F¯ = F◦ \ F (V \ S), so every face in F¯ contains only vertices of S. We will show that

$$|\bar{F}|\pi < \sum\_{v \in S} \theta(v). \tag{3.7}$$

If (3.7) holds, then we can add the negative quantity |F¯|π − <sup>v</sup>∈<sup>S</sup> θ (v) to the right side of (3.6), obtaining |F◦|π − <sup>v</sup>∈<sup>V</sup> θ (v) <sup>=</sup> (2n−5)π <sup>−</sup>(2n−5)π <sup>=</sup> 0. It follows that (3.6) is strictly positive, proving the claim. Thus it suffices to show (3.7).

In the rest of the proof, we fix an embedding of G in the plane with (v1, v2, v3) as the outer face. Let G[S] be the subgraph of G induced by S. Partition S into equivalence classes, S = S<sup>1</sup> ∪···∪Sk, where two vertices are equivalent if they are in the same connected component of G[S]. Then G[S] = G[S1]∪···∪ G[Sk]. Let F¯ <sup>j</sup> be the set of faces in F¯ that appear as faces of G[Sj ], so that we have the disjoint union F¯ = F¯ <sup>1</sup> ∪···∪ F¯ k .

Since S is nonempty, it is enough to show that for all 1 ≤ j ≤ k,

$$|\bar{F}\_f|\pi < \sum\_{\upsilon \in S\_f} \theta(\upsilon). \tag{3.8}$$

Let mj and fj denote the number of edges and faces, respectively, of G[Sj ]. Observe that |F¯ <sup>j</sup> | ≤ fj − 1. If |F¯ <sup>j</sup> | = 0, then (3.8) is trivial. If |F¯ <sup>j</sup> | ≥ 1, then G[Sj ] has at least one inner face, and since it is a simple graph, every face must have degree at least 3. (The degree of a face is the number of directed edges that make up its boundary.) Because the sum of the degrees of all the faces equals twice the number of edges, we have 2mj ≥ 3fj . Euler's formula now gives

$$|S\_j| + f\_j - 2 = m\_j \ge \frac{3}{2} f\_j, j$$

and hence fj ≤ 2|Sj | − 4. Thus, the left side of (3.8) satisfies

$$|\bar{F}\_j|\pi \le (\mathcal{Z}|S\_j| - \mathcal{S})\pi.$$

If Sj contains all of v1, v2, v3, then the right side of (3.8) is

$$
\pi(\theta\_1 + \theta\_2 + \theta\_3 + (|\mathcal{S}\_f| - \mathfrak{Z}) \cdot 2\pi = (2|\mathcal{S}\_f| - \mathfrak{Z})\pi.
$$

Otherwise, at least one of the θi is replaced by 2π and so the right side of (3.8) is strictly greater than the left side. In fact, (3.8) holds except when v1, v2, v<sup>3</sup> ∈ Sj and |F¯ <sup>j</sup> | = fj − 1 = 2|Sj | − 5. We now show that this situation cannot occur.

The equality |F¯ <sup>j</sup> | = fj − 1 means that every inner face of G[Sj ] is an element of F¯ <sup>j</sup> and therefore a face of G. Since v1, v2, v<sup>3</sup> ∈ Sj , the outer face of G[Sj ] is (v1, v2, v3), which is the same as the outer face of G. So, every face of G[Sj ] is also a face of G. But this is impossible: if we choose any v ∈ V \ S, then v must lie in some face of G[Sj ], which then cannot be a face of G. Therefore, it cannot be true that v1, v2, v<sup>3</sup> ∈ Sj and also |F¯ <sup>j</sup> | = fj − 1, so we conclude that (3.8) always holds. 

We now analyse the algorithm. Let λ ∈ (0, 1) be the one guaranteed by Claim 3.10, and set **r** = **r**¯λ.

$$\text{Claim 3.12 } \mathcal{E'} \le \mathcal{E} \left(1 - \frac{1}{2n^3} \right).$$

*Proof* As depicted in Fig. 3.5, define

$$t = \min\_{v \in S} \delta'(v) = \max\_{v \notin S} \delta'(v).$$

By (3.4) we have that <sup>n</sup> <sup>i</sup>=<sup>1</sup> δ(vi) <sup>=</sup> <sup>n</sup> <sup>i</sup>=<sup>1</sup> <sup>δ</sup> (vi) = 0, hence

$$\mathcal{E} - \mathcal{E}' = \sum\_{l=1}^{n} \delta(v\_l)^2 - \sum\_{l=1}^{n} \delta'(v\_l)^2 = \sum\_{l=1}^{n} (\delta(v\_l) - \delta'(v\_l))^2 + 2\sum\_{l=1}^{n} (t - \delta'(v\_l))(\delta'(v\_l) - \delta(v\_l)).$$

If v ∈ S, then t ≤ δ (v) ≤ δ(v) and if v /∈ S, then t ≥ δ (v) ≥ δ(v). Thus, in both cases (t −δ (v))(δ (v)−δ(v)) ≥ 0. Taking u ∈ S and v /∈ S with δ (u) = δ (v) = t, we have that

$$
\mathcal{E} - \mathcal{E}' \ge \left(\delta(\mu) - t\right)^2 + \left(\delta(\upsilon) - t\right)^2 \ge \frac{\left(\delta(\mu) - \delta(\upsilon)\right)^2}{2} \ge \frac{\text{gap}\_{\delta}(\mathcal{S})^2}{2}.
$$

Since gapδ(S) was chosen to be the maximal gap we may bound,

$$\text{gap}\_{\delta}(S) \ge \frac{1}{n} \left( \max\_{v \in V} \delta(v) - \min\_{v \in V} \delta(v) \right).$$

For every v ∈ V ,

$$\max\_{w \in V} \delta(w) - \min\_{w \in V} \delta(w) \ge |\delta(v)|,$$

and thus

$$n\left(\max\_{\upsilon\in V}\delta(\upsilon) - \min\_{\upsilon\in V}\delta(\upsilon)\right)^2 \ge \sum\_{l=1}^n \delta(\upsilon\_l)^2 = \mathcal{E}\dots$$

Hence

$$
\mathcal{E} - \mathcal{E}' \ge \frac{1}{2n^2} \left( \max\_{\upsilon \in V} \delta(\upsilon) - \min\_{\upsilon \in V} \delta(\upsilon) \right)^2 \ge \frac{1}{2n^2} \cdot \frac{\mathcal{E}}{n},
$$

and we conclude that

$$\mathcal{E}' \le \mathcal{E} \left( 1 - \frac{1}{2n^3} \right). \tag{7}$$

Write *<sup>E</sup>*(t ) <sup>=</sup> *<sup>E</sup>***r**(t). By iterating the described algorithm, we obtain from Claim 3.12 that

$$\mathcal{E}^{(t)} \le \mathcal{E}^{(0)} \left(1 - \frac{1}{2n^3}\right)^t \longrightarrow 0 \qquad \text{as } t \to \infty.$$

By our normalization **<sup>r</sup>**(t ) <sup>1</sup> = 1. Thus, by compactness, there exists a subsequence {tk} and a vector **<sup>r</sup>**<sup>∞</sup> such that **<sup>r</sup>**(tk) <sup>→</sup> **<sup>r</sup>**<sup>∞</sup> as <sup>k</sup> → ∞. From continuity of *<sup>E</sup>* we have that *E* (**r**∞) = 0, meaning that (3.2) is satisfied. For **r**<sup>∞</sup> to be feasible as a vector of radii, we also have to argue that it is positive (the fact that no coordinates are ∞ follows since ||**r**∞||<sup>1</sup> = 1).

*Claim 3.13* **r**∞ <sup>i</sup> > 0 for every i.

*Proof* Let S = {vi ∈ V : **r**<sup>∞</sup> <sup>i</sup> > 0}. Because of the normalization of **r**, we know that S is nonempty. Assume for contradiction that S - V . We repeat the exact same argument used in the proof of Claim 3.11 showing first by case analysis that

$$\lim\_{t \to \infty} \sum\_{\upsilon \notin S} \sigma\_{\mathbf{r}^{(l)}}(\upsilon) = |F(V \backslash S)|\pi$$

and then deducing that

$$\lim\_{\mathfrak{r}\to\infty} \sum\_{\upsilon\notin S} \delta\_{\mathbf{r}^{(\mathfrak{r})}}(\upsilon) > 0.$$

This contradicts that limt→∞ *<sup>E</sup>*(t ) <sup>=</sup> 0, so we conclude that <sup>S</sup> <sup>=</sup> <sup>V</sup> . 

#### *Proof of Theorem 3.6, Step 2: Drawing the Circle Packing Described by* **r∞**

Given the vector of radii **r**∞ satisfying (3.2), we now show that the corresponding circle packing can be drawn uniquely up to translations and rotations. In fact, we provide a slightly more general statement which is due to Ori Gurel-Gurevich and Ohad Feldheim [personal communications, 2018].

Let G = (V , E) be a finite planar triangulation on vertex set {v1,...,vn} and assume that {v1, v2, v3} is the outer face. A vector of positive real numbers *-* = {e}e∈<sup>E</sup> indexed by the edge set E is called *feasible* if for any face enclosed by edges e1, e2, e3, the lengths e<sup>1</sup> , e<sup>2</sup> , e<sup>3</sup> can be made to form a triangle. In other words, these lengths satisfy three triangle inequalities,

$$\ell\_{\mathfrak{e}\_{\mathbb{I}}} + \ell\_{\mathfrak{e}\_{\mathbb{J}}} \succ \ell\_{\mathfrak{e}\_{\mathbb{K}}} \qquad \{i, j, k\} = \{1, 2, 3\} \dots$$

Given a feasible edge length vector  we may again use the cosine formula to compute, for each face f , the angle at a vertex of the triangle formed by the three corresponding edge lengths. We denote these angles, as before, by α*-* <sup>f</sup> (v) where v is a vertex of f . Similarly, we define

$$\sigma\_{\ell}(v) = \sum\_{f \in F(v)} \alpha\_f^{\ell}(v)$$

to be the sum of angles at a vertex v.

**Theorem 3.14** *Let* G *be a finite triangulation and a feasible vector of edge lengths. Assume that* σ*-*(v) = 2π *for any internal vertex* v*. Then there is a drawing of* G *in the plane so that each edge* e *is drawn as a straight line segment of length* e *and no two edges cross. Furthermore, this drawing is unique up to translations and rotations.*

It is easy to use the theorem above to draw the circle packing given the radii vector **r**∞ satisfying (3.2). Indeed, given **r**∞ we set  by putting e = **r**<sup>∞</sup> <sup>i</sup> + **r**<sup>∞</sup> <sup>j</sup> for any edge e = {vi, vj } of the graph. Condition (3.2) implies that  is feasible. We now apply Theorem 3.14 and obtain the guaranteed drawing and draw a circle Ci of radii **r**∞ <sup>i</sup> around vi for all i. Theorem 3.14 guarantees that for any edge {vi, vk } the distance between vi, vj is precisely **r**<sup>∞</sup> <sup>i</sup> + **r**<sup>∞</sup> <sup>j</sup> and thus Ci and Cj are tangent. Conversely, assume that vi, vj do not form an edge. To each vertex v let Av be the union of triangles touching v, each triangle is the space bounded by a face touching v in the drawing of Theorem 3.14. Since G is a triangulation and vi and vj are not adjacent we learn that Avi and Avj have disjoint interiors. Furthermore, Ci ⊂ Int(Avi) since the straight lines emanating from vi have length larger than **r**<sup>∞</sup> <sup>i</sup> . By the same token Cj ⊂ Int(Avj ) and we conclude that Ci and Cj are not tangent.

Lastly, we note that by (3.2) the outer boundary of the polygon we drew is a triangle with angles θ1, θ2, θ<sup>3</sup> and hence (r1, r2, r3) is a positive multiple of (ρ1, ρ2, ρ3). Step 2 of the proof of Theorem 3.5 is now concluded.

*Proof of Theorem 3.14* We prove this by induction on the number of vertices n. The base case n = 3 is trivial since the feasibility of  guarantees that the edge lengths of the three edges of the outer face can form a triangle. Any two triangles with the same edge lengths can be rotated and translated to be identical, so the uniqueness statement holds for n = 3.

Assume now that n > 3 so that there exists an internal vertex v. Denote by v1,...,vm the neighbors of v ordered clockwise. We begin by placing v at the origin and drawing all the faces to which v belongs, see Fig. 3.6, left. That is, we draw the edge {v, v1} as a straight line interval of length {v,v1} on the positive xaxis emanating from the origin and proceed iteratively: for each 1 < i ≤ m we draw the edge {v, vi} as a straight line interval of length {v,vi} emanating from the origin (v) at a clockwise angle of α*-* <sup>f</sup> (v) from the previous drawn line segment of {v, vi−1}, where f = {v, vi−<sup>1</sup>, vi}. This determines the location of v1,...,vm in the plane and allows us to "complete" the triangles by drawing the straight line

**Fig. 3.6** On the left, we first draw the polygon surrounding v. On the right, we then erase v and the edge emanating from it, replacing it with diagonals that triangulate the polygon while recording the lengths of the diagonals in *-* . The latter is the input to the induction hypothesis

segments connecting vi to vi+1, each of length {vi,vi+1} where 1 ≤ i ≤ m (where vm+<sup>1</sup> = v1). Denote these edges by e1,...,em.

Since σ*-*(v) = 2π we learn that these m triangles have disjoint interiors and that the edges e1,...,em form a closed polygon containing the origin in its interior. It is a classical fact [64] that every closed polygon can be triangulated by drawing some diagonals as straight line segments in the interior of the polygon. We fix such a choice of diagonals and use it to form a new graph G on n − 1 vertices and |E(G)| − 3 edges by erasing v and the m edges emanating from it and adding the new m−3 edges corresponding to the diagonals we added. Furthermore, we generate a new edge length vector  corresponding to G by assigning the new edges lengths corresponding to the Euclidean length of the drawn diagonals and leaving the other edge lengths unchanged. See Fig. 3.6, right.

It is clear that  is feasible and that the angle sum at each internal vertex of G is 2π. Therefore we may apply the induction hypothesis and draw the graph G according to the edge lengths *-* . This drawing is unique up to translations and rotations by induction. Note that in this drawing of G , the polygon corresponding to e1,...,em must be the exact same polygon as before, up to translations and rotations, since it has the same edge lengths and the same angles between its edges. Since it is the same polygon, we can now erase the diagonals in this drawing and place a new vertex in the same relative location where we drew v previously, along with the straight line segments connecting it to v1,...,vm. Thus we have obtained the desired drawing of G. The uniqueness up to translations and rotations of this drawing follows from the uniqueness of the drawing of G and the fact that the location of v is uniquely determined in that drawing. 

#### *Proof of Theorem 3.6, Step 3: Uniqueness*

**Theorem 3.15 (Uniqueness of Circle Packing)** *Given a simple finite triangulation with outer face* v1, v2, v<sup>3</sup> *and three radii* ρ1, ρ2, ρ3*, the circle packing with* Cv<sup>1</sup> , Cv<sup>2</sup> , Cv<sup>3</sup> *having radii* ρ1, ρ2, ρ<sup>3</sup> *is unique up to translations and rotations.*

*Proof* We have already seen in step 2 that given the radii vector **r** the drawing we obtain is unique up to translations and rotations. Thus, we only need to show the uniqueness of **r** given ρ1, ρ2, ρ3.

To that aim, suppose that **r**<sup>a</sup> and **r**<sup>b</sup> are two vectors satisfying (3.2). Since the outer face in both vectors correspond to a triangle of angles θ1, θ2, θ<sup>3</sup> we may rescale so that **r**<sup>a</sup> <sup>i</sup> <sup>=</sup> **<sup>r</sup>**<sup>b</sup> <sup>i</sup> = ρi for i = 1, 2, 3. After this rescaling, assume by contradiction that **<sup>r</sup>**<sup>a</sup> = **<sup>r</sup>**<sup>b</sup> and let <sup>v</sup> be the interior vertex which maximizes **<sup>r</sup>**<sup>a</sup> v/**r**<sup>b</sup> <sup>v</sup>. We can assume without loss of generality that this quantity is strictly larger than 1, as otherwise we can swap **r**<sup>a</sup> and **r**b.

Now we claim that for each <sup>f</sup> <sup>=</sup> (v, u1, u2) <sup>∈</sup> F (v), we have <sup>α</sup>**r**<sup>a</sup> <sup>f</sup> (v) <sup>≤</sup> <sup>α</sup>**r**<sup>b</sup> <sup>f</sup> (v), with equality if and only if the ratios **r**<sup>a</sup> ui /**r**<sup>b</sup> ui , for <sup>i</sup> <sup>=</sup> <sup>1</sup>, 2, are both equal to **<sup>r</sup>**<sup>a</sup> <sup>v</sup> /**r**<sup>b</sup> v. This is a direct consequence of Observation 3.8. Indeed, scale all the radii in **r**<sup>b</sup> by a factor of **r**<sup>a</sup> v/**r**<sup>b</sup> <sup>v</sup> to get a new vector **r** such that **r**<sup>a</sup> <sup>v</sup> = **r** <sup>v</sup> and **r**<sup>a</sup> <sup>u</sup> ≤ **r** <sup>u</sup> for all u = v. The second bullet point in Observation 3.8 implies that α**r**<sup>a</sup> <sup>f</sup> (v) <sup>≤</sup> <sup>α</sup>**r** <sup>f</sup> (v) <sup>=</sup> <sup>α</sup>**r**<sup>b</sup> <sup>f</sup> (v). As well, if either **r**<sup>a</sup> <sup>u</sup><sup>1</sup> < **r** <sup>u</sup><sup>1</sup> or **<sup>r</sup>**<sup>a</sup> <sup>u</sup><sup>2</sup> < **r** <sup>u</sup><sup>2</sup> , then the cosine formula yields the strict inequality α**r**<sup>a</sup> <sup>f</sup> (v) < α**r** <sup>f</sup> (v). Thus, <sup>α</sup>**r**<sup>a</sup> <sup>f</sup> (v) <sup>=</sup> <sup>α</sup>**r**<sup>b</sup> <sup>f</sup> (v) only if **<sup>r</sup>**<sup>a</sup> ui /**r**<sup>b</sup> ui <sup>=</sup> **<sup>r</sup>**<sup>a</sup> <sup>v</sup> /**r**<sup>b</sup> <sup>v</sup> for i = 1, 2.

Now, since α**r**<sup>a</sup> <sup>f</sup> (v) <sup>≤</sup> <sup>α</sup>**r**<sup>b</sup> <sup>f</sup> (v) for each f ∈ F (v), while σ**r**<sup>a</sup> (v) = σ**r**<sup>b</sup> (v) = 2π, the equality α**r**<sup>a</sup> <sup>f</sup> (v) <sup>=</sup> <sup>α</sup>**r**<sup>b</sup> <sup>f</sup> (v) must hold for each f . Therefore, each neighbor u of v satisfies **r**<sup>a</sup> u/**r**<sup>b</sup> <sup>u</sup> <sup>=</sup> **<sup>r</sup>**<sup>a</sup> v/**r**<sup>b</sup> <sup>v</sup>. Because the graph is connected, the ratio **r**<sup>a</sup> u/**r**<sup>b</sup> <sup>u</sup> must be constant for all vertices <sup>u</sup> <sup>∈</sup> V (G). But this contradicts that **<sup>r</sup>**<sup>a</sup> <sup>v</sup> /**r**<sup>b</sup> <sup>v</sup> > 1 while **r**a vi /**r**<sup>b</sup> vi <sup>=</sup> 1 for <sup>i</sup> <sup>=</sup> <sup>1</sup>, <sup>2</sup>, 3. We conclude that **<sup>r</sup>**<sup>a</sup> <sup>=</sup> **<sup>r</sup>**b. 

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# **Chapter 4 Parabolic and Hyperbolic Packings**

#### **4.1 Infinite Planar Maps**

In this chapter we discuss countably infinite connected simple graphs that are **locally finite**, that is, the vertex degrees are finite. In a similar fashion to the previous chapter, an infinite planar graph is a connected infinite graph such that there exists a drawing of it in the plane. We recall that a *drawing* is a correspondence sending vertices to points of R<sup>2</sup> and edges to continuous curves between the corresponding vertices such that no two edges cross. An **infinite planar map** is an infinite planar graph equipped with a set of cyclic permutations {σv : v ∈ V } of the neighbors of each vertex v, such that there exists a drawing of the graph which respects these permutations, that is, the clockwise order of edges emanating from a vertex v coincides with σv.

Unlike the finite case, one cannot define faces as the connected components of the plane with the edges removed since the drawing may have a complicated set of accumulation points. This is the reason that we have defined faces in Sect. 3.1 combinatorially, that is, based solely on the edge set and the cyclic permutation structure. This definition makes sense in both the finite and infinite case. In the latter case we may have infinite faces.

A (finite or infinite) planar map is a **triangulation** if each of its faces has exactly 3 edges. Given a drawing of a triangulation, the Jordan curve theorem implies that the edges of each face bound a connected component of the plane minus the edges. We will often refer to the faces as these connected components. A triangulation is called a **plane triangulation** if there exists a drawing of it such that every point of the plane is contained in either a face or an edge and any compact subset of the plane intersects at most finitely many edges and vertices. The term **disk triangulation** is also used in the literature and means the same with the unit disk taking the place of the plane in the previous definition. Of course these two definitions are equivalent since the plane and the open disk are homeomorphic. For example, take the product of the complete graph K<sup>3</sup> on 3 vertices with an infinite ray N and add a diagonal edge

A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes in Mathematics 2243, https://doi.org/10.1007/978-3-030-27968-4\_4

in each face that has 4 edges; this is a plane triangulation. However, the product of K<sup>3</sup> with a bi-infinite ray Z together with the same diagonals is a triangulation but not a plane triangulation, since it cannot be drawn in the plane without an accumulation point.

It turns out that there is a combinatorial criterion for a triangulation to be a plane/disk triangulation. We say that an infinite graph is **one-ended** if the removal of any finite set of its vertices leaves exactly one infinite connected component.

**Lemma 4.1** *An infinite triangulation is a plane triangulation if and only if it is one-ended.*

*Proof* Suppose G = (V , E) is a plane triangulation and consider a drawing of the graph with no accumulation points in the plane such that every point of the plane belongs to either an edge or a face. Let A ⊆ V be a finite set of vertices and take <sup>B</sup> <sup>⊂</sup> <sup>R</sup><sup>2</sup> to be a ball around the origin which contains every vertex of <sup>A</sup>, every edge touching a vertex of A and every face incident to such an edge. Let u = v be two vertices drawn outside of <sup>B</sup> and take a continuous curve <sup>γ</sup> between them in <sup>R</sup><sup>2</sup> \ <sup>B</sup>. By definition of B, this path only touches faces and edges that are not incident to the vertices of A and hence one can trace a discrete path from u to v in the graph that "follows" γ and avoids A. Since B intersects only finitely many edges and vertices, we learn that G \ A has a unique infinite component.

Conversely, assume now that G is one-ended and consider a drawing of G in the plane. By the stereographic projection we project the drawing to the unit sphere S<sup>2</sup> in <sup>R</sup>3. Denote by *<sup>I</sup>* the complement in <sup>S</sup><sup>2</sup> of the union of all faces and edges. Since G is an infinite triangulation this union is an open set, hence *I* is a closed set and its boundary ∂*I* is precisely the set of accumulation points of the drawing. Since *I* is closed, each connected component of *I* must be closed as well and hence contain at least one accumulation point. Since G is one-ended *I* cannot have more than one connected component, since otherwise we would be able to separate the two components by a finite set of edges and obtain two infinite connected components. Now choose a point p ∈ *I* and rotate the sphere so that p is the north pole. Project back the rotated sphere to the plane and consider the drawing in the plane. In this drawing the union of all faces and edges must be a simply connected set. By the Riemann mapping theorem this set is homeomorphic to the whole plane, and we deduce that the triangulation is a plane triangulation. 

#### **4.2 The Ring Lemma and Infinite Circle Packings**

The circle packing theorem Theorem 3.5 is stated for finite planar maps. However, it is not hard to argue that any infinite map also has a circle packing. To this aim we will prove what is known as Rodin and Sullivan's *Ring Lemma* [70]; we will use it many times throughout this book. Given circles C0, C1,...,CM with disjoint interiors, we say that C1,...,CM completely surround C<sup>0</sup> if they are all tangent to C<sup>0</sup> and Ci is tangent to Ci+<sup>1</sup> for i = 1,...,M (where CM+<sup>1</sup> is set to be C1).

**Lemma 4.2 (Ring Lemma, Rodin and Sullivan [70])** *For every integer* M > 0 *there exists* A > 0 *such that if* C<sup>0</sup> *is a circle completely surrounded by* M *circles* C1,...,CM *, and* ri *is the radius of* Ci *for every* i = 0, 1,...,M*, then* r0/ri ≤ A *for every* i = 1,...,M*.*

*Proof* We may scale the picture so that r<sup>0</sup> = 1. Assume that the radius of C<sup>2</sup> is small and consider the circles C<sup>1</sup> and C<sup>3</sup> to its left and right. It cannot be that both C<sup>1</sup> and C<sup>3</sup> have large radii compared to C<sup>2</sup> since in this case they will intersect; see Fig. 4.1. Hence, one of them has to be small as well. Assume without loss of generality that it is C3. By similar reasoning, one of C<sup>1</sup> and C<sup>4</sup> has to be small. We continue this argument this way and get a path of circles of small radii; thus, for the circles C1,...,CM to completely surround C<sup>0</sup> we learn that M must be large. 

For a circle packing P and a vertex v, denote by Cv the circle corresponding to v, by cent(v) the center of that circle, and by rad(v) its radius. We write G(P ) for the tangency graph of the packing P, that is, the graph in which each vertex is a circle of P and two such circles form an edge when they are tangent.

*Claim 4.3* Let G be an infinite simple planar map. Then there exists a circle packing P such that G(P ) is isomorphic to G as planar maps.

*Proof* If G is not a triangulation, then it is always possible to add in each face new vertices and edges touching them so the resulting graph is a planar triangulation (in an infinite face we have to put infinitely many vertices). After circle packing this new graph, we can remove all the circles corresponding to the added vertices and remain with a circle packing of G. Thus, we may assume without loss of generality that G is a triangulation.

Fix a vertex x, and let Gn be the graph distance ball of radius n around x. Apply the circle packing theorem to Gn to obtain a packing Pn, and scale and translate it so that rad(x) = 1 and cent(x) is the origin.

Consider a neighbor y of x. By the Ring Lemma (Lemma 4.2), there exists a constant <sup>A</sup> <sup>=</sup> A(x, y) > 0 such that <sup>A</sup>−<sup>1</sup> <sup>≤</sup> rad(y) <sup>≤</sup> <sup>A</sup>. By compactness there exists a subsequence of packings Pnk for which radnk (y) and centnk (y) both converge. By taking further subsequences for the rest of x's neighbors, and then for the rest of the graph's vertices, it follows by a diagonalization argument that there exists a subsequence such that the radii and centers of all vertices converge. The limiting packing P<sup>∞</sup> satisfies that G(P∞) is isomorphic to G. 

#### **4.3 Statement of the He–Schramm Theorem**

Given a circle packing P of a triangulation G, we define the **carrier** of P, denoted Carrier(P ), to be the union of the closed discs bounded by the circles of P together with the spaces bounded between any three circles that form a face (i.e., the interstices). When P is a circle packing of an infinite one-ended triangulation, the argument in Lemma 4.1 shows that Carrier(P ) is simply connected.

We say that <sup>G</sup> **is circle packed in** <sup>R</sup><sup>2</sup> when Carrier(P ) <sup>=</sup> <sup>R</sup>2. Denote by <sup>U</sup> the disk {<sup>z</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup> : |z<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>}; we say that <sup>G</sup> **is circle packed in** <sup>U</sup> when Carrier(P ) <sup>=</sup> <sup>U</sup>. See Fig. 4.2.

Let G be a plane triangulation. Then G can be drawn in the plane R<sup>2</sup> or alternatively in the disk U (since they are homeomorphic), but can it be *circle packed* both in R<sup>2</sup> and in U? A celebrated theorem of He and Schramm [40] states that this cannot be done: each plane triangulation can be circle packed in either the plane or the disk, but not both. In fact, the combinatorial property of G that determines on which side of the dichotomy we are is the recurrence or transience of the simple random walk on G (assuming also that G has bounded degrees, that is, supx∈V (G) deg(x) < ∞). This is the content of the He–Schramm theorem, which we are now ready to state.

**Theorem 4.4 (He and Schramm [40])** *Let* G *be an infinite simple plane triangulation with bounded degrees.*


**Fig. 4.2** Two circle packings with carriers R<sup>2</sup> (left) and U (right)

*3. If* <sup>P</sup> *is a circle packing of* <sup>G</sup> *with* Carrier(P ) <sup>=</sup> <sup>R</sup>2*, then* <sup>G</sup> *is recurrent. 4. If* <sup>P</sup> *is a circle packing of* <sup>G</sup> *with* Carrier(P ) <sup>=</sup> <sup>U</sup>*, then* <sup>G</sup> *is transient.*

*Remark 4.5* Schramm [73] proved that a circle packing P of a triangulation G(P ) with Carrier(P ) <sup>=</sup> <sup>R</sup><sup>2</sup> is uniquely determined up to dilations, rotations and translations. If Carrier(P ) <sup>=</sup> <sup>U</sup> the same holds up to Möbius transformations of <sup>U</sup> onto itself (see also [37]). Hence the packings guaranteed to exist in Theorem 4.4 (1) and Theorem 4.4 (2) are unique in this sense.

**Corollary 4.6** *Any bounded degree plane triangulation can be circle-packed in* R<sup>2</sup> *or* U*, but not both.*

*Remark 4.7* In fact, it is proved in [40] that the corollary above holds without the assumption of bounded degree. Furthermore, in [40] Theorem 4.4 (1) and Theorem 4.4 (4) are proved without the bounded degrees assumption, but the other two statements require this assumption.

The following example demonstrates why the bounded degree condition is necessary for Theorem 4.4 (2) and Theorem 4.4 (3).

*Example 4.8* Let P be a triangular lattice circle packing (as in Fig. 4.3), and let C0, C1, C2,... be an infinite horizontal path of circles in P going (say) to the right. In the upper face shared by Cn and Cn+1, draw 2<sup>n</sup> circles which form a vertical path and each of them tangent both to Cn and Cn+1; the last circle of these is also tangent to the upper neighbor of Cn and Cn+1. See Fig. 4.3.

The resulting graph is a plane triangulation and the carrier of the packing is R2. However, it is an easy exercise to verify that the tangency graph of this circle packing is transient.

In the rest of this chapter we prove Theorem 4.4. We begin by proving parts 3 and 4, in which a circle packing is given and we use its geometry to estimate

**Fig. 4.3** Unbounded degree transient triangulation circle packed in R<sup>2</sup>

certain effective resistances. Afterwards we prove parts 1 and 2, in which we use the electrical estimates to deduce facts about the geometry of the circle packing.

#### **4.4 Proof of the He–Schramm Theorem**

#### *Proof of Theorem 4.4 (3)*

Denote the circle packing P = {Cv}v∈<sup>V</sup> where V is the vertex set of G and Cv denotes the circle corresponding to the vertex v. Write Δ for the maximum degree of G and fix a vertex v0. By scaling and translating we may assume that Cv<sup>0</sup> is a radius 1 circle around the origin. For a real number R > 0, let VR = VB(**<sup>0</sup>**,R) denote set of vertices v for which cent(v) is in the Euclidean ball of radius R around the origin.

**Lemma 4.9** *There exist* C = C(Δ) > 1 *and* c = c(Δ) > 0 *such that for every* R ≥ 1 *we have*


*Proof* We begin with part (i). For every v ∈ VR it holds that rad(v) ≤ R since Cv<sup>0</sup> is centered at the origin. By the Ring Lemma (Lemma 4.2), there exists A = A(Δ) such that rad(u) ≤ AR for every u ∼ v , and therefore | cent(u)| ≤ (A + 2)R. Hence (i) holds with C = A + 2.

To prove part (ii) we define

$$h(v) = \begin{cases} 0 & v \in V\_R, \\ 1 & v \in V \nmid V\_{CR}, \\ \frac{|\text{cent}(v)| - R}{(C - 1)R} & \text{otherwise.} \end{cases}$$

Recall from Lemma 2.32 that *<sup>R</sup>*eff (VR <sup>↔</sup> <sup>V</sup> \ VCR) <sup>≥</sup> *<sup>E</sup>*(h)−1. By the triangle inequality, for an edge {x,y} with both endpoints in VCR \ VR we have

$$|h(\mathbf{x}) - h(\mathbf{y})| \le \frac{|\mathsf{cent}(\mathbf{x}) - \mathsf{cent}(\mathbf{y})|}{(C - 1)R} = \frac{\mathsf{rad}(\mathbf{x}) + \mathsf{rad}(\mathbf{y})}{(C - 1)R},$$

and it is straightforward to check that the same bound holds also when one of the edge's endpoints is in VR or V \ VCR. Thus, using the Ring Lemma's (Lemma 4.2) constant A = A(Δ) from part (i),

$$\mathcal{E}(h) \le \sum\_{\boldsymbol{x} \in V\_{CR} \backslash V\_R} \sum\_{\boldsymbol{Y} : \boldsymbol{Y} \sim \boldsymbol{X}} \frac{((\boldsymbol{A} + \boldsymbol{1})\operatorname{rad}(\boldsymbol{x}))^2}{(\boldsymbol{C} - \boldsymbol{1})^2 R^2} \le \frac{\Delta (\boldsymbol{A} + \boldsymbol{1})^2}{\pi (\boldsymbol{C} - \boldsymbol{1})^2 R^2} \cdot \sum\_{\boldsymbol{x} \in V\_{CR \mid V\_R}} \operatorname{area}(\mathcal{C}\_{\boldsymbol{x}}),$$

where area(Cx ) is the area that Cx encloses (that is, π rad(x)<sup>2</sup> ). We have that <sup>x</sup> area(Cx ) <sup>≤</sup> area(B(**0**, <sup>2</sup>CR)) <sup>=</sup> <sup>4</sup>πC2R2, hence if <sup>C</sup> <sup>=</sup> <sup>A</sup> <sup>+</sup> 2, then

$$
\mathcal{E}(h) \le 4\Delta C^2,
$$

and the result follows for <sup>c</sup> <sup>=</sup> (4ΔC2)−1. 

*Proof of Theorem 4.4 (3)* Consider the unit current flow I from v<sup>0</sup> to ∞ and fix any R ≥ 1. Restricting this flow to the edges which have at least one endpoint in the annulus VCR \ VR gives a unit flow from VR to V \ VCR, by part (i) of Lemma 4.9. Hence, by part (ii) of that lemma and by Thomson's principle (Theorem 2.28), the energy contributed to *E*(I ) from these edges is at least c. In the same manner, the edges which have at least one endpoint in the annulus VC2k+1<sup>R</sup> \ VC2kR contribute at least c to *E*(I ). Part (i) of Lemma 4.9 implies that all these edge sets are disjoint, hence *E*(I ) = ∞ and we learn that G is recurrent (Corollary 2.39). 

#### *Proof of Theorem 4.4 (4)*

We will use the given circle packing of G to create a random path to infinity with finite energy. This gives transience by Claim 2.46. This proof strategy is similar to that of Theorem 2.47.

*Proof of Theorem 4.4 (4)* Let v<sup>0</sup> be a fixed vertex of the graph, and apply a Möbius transformation to make the circle of P corresponding to v<sup>0</sup> be centered at the origin **0**. We now use Claim 2.46 to construct a flow θ from v<sup>0</sup> to ∞ by choosing a uniform random point **p** on ∂U, taking the straight line from **0** to **p** and considering the set of all circles in the packing P that intersect this line in the order that they are visited; this set forms an infinite simple path in the graph which starts at v0.

To bound the energy of the flow, we claim that there exists some constant C (which may depend on the graph G and the packing P) such that the probability that the random path uses the vertex v is bounded above by C rad(v). Indeed, since there are only finitely many vertices with centers at distance at most 1/2 from **0**, we may assume that the center of v is of distance at least 1/2 from **0**. In this case, in order for v to be included in the random path the circle of v must intersect the line between **0** and **p**. By the Ring Lemma (Lemma 4.2) the neighbors of v have circles of radii comparable to rad(v) and so the probability of the line touching them is at most C rad(v). Since the vertex degree is bounded by Δ and <sup>v</sup>∈<sup>V</sup> <sup>π</sup> rad(v)<sup>2</sup> is at most the area of U, we find that

$$\mathcal{E}(\theta) \le C\Delta \sum\_{v \in V} \text{rad}(v)^2 \le C\Delta\ \therefore$$

Hence G is transient by Corollary 2.39 

#### *Proof of Theorem 4.4 (1)*

We apply Claim 4.3 to obtain a circle packing P of G and prove that Carrier(P ) = R2. Fix some vertex v and rescale and translate so that P (v) is the unit circle ∂U. Assume by contradiction that Carrier(P ) = <sup>R</sup><sup>2</sup> and let <sup>p</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup> \ Carrier(P ) be a point not in the carrier. Rotate the packing so that p = R for some real number R > 1. Let U ∈ [−1, 1] and consider the circle CU = {z : |z − p| = R − U}. We traverse CU from the point U counterclockwise and consider all the circles of P which intersect CU . These circles form a simple path in the graph G starting from v. Since Carrier(P ) is simply connected by Lemma 4.1 and p ∈ Carrier(P ) it cannot be that CU ⊂ Carrier(P ). Thus, as we traverse CU counterclockwise we must hit the boundary of Carrier(P ). We conclude that the path in G we obtained in this manner is an infinite simple path starting at v.

We now let U be a uniform random variable in [−1, 1] and let μ denote the probability measure on random infinite paths starting at v we obtained as described above. Let θ be the flow induced by μ as in Claim 2.46. We wish to bound the energy *E*(θ ). Consider a vertex w ∈ G and its corresponding circle Cw and let B be the Euclidean ball of radius R + 1 around p. If Cw does not intersect B, it cannot be included in the random path by our construction. If it does intersect this ball, then the probability that the random path intersects it is bounded above by its radius. Thus as in the proof of Theorem 4.4 (4),

$$\mathcal{E}(\theta) \le C\Delta \sum\_{w:C\_w \cap B \ne \emptyset} \text{rad}(w)^2,$$

where Δ is the maximal degree of G and we have used the Ring Lemma (Lemma 4.2). We learn that *E*(θ ) is bounded above by a constant multiple of the area of all circles of P that intersect B. Since p ∈ Carrier(P ), by the Ring Lemma (Lemma 4.2), any circle of P that intersects B cannot have radius more than AR for some large A ≥ R (since otherwise, all the circles surrounding this vertex will have radius more than R + 1, contradicting the fact that p ∈ Carrier(P )). We learn that all the circles counted in the sum above are contained in the Euclidean ball of radius (A + 1)R + 1 around p. Since these circles has disjoint interiors, the sum of their area is bounded above by the area of the Euclidean ball above. We conclude that *E*(θ ) < ∞, hence G is transient by Corollary 2.39 and we have reached a contradiction. 

#### *Proof of Theorem 4.4 (2)*

We will use the following simple corollary of the circle packing theorem, Theorem 3.5. A **finite triangulation with boundary** is a finite connected simple planar map in which all faces are triangles except for a distinguished outer face whose boundary is a simple cycle.

*Claim 4.10* Let G be a finite triangulation with boundary. Then, there is a circle packing P of G such that all circles of the outer face are internally tangent to ∂U and all other circles of P are contained in U.

*Proof* Denote by v1,...,vm the vertices of the outer face ordered according to the cycle they form. Add a new vertex v<sup>∗</sup> to the graph and connect it to v1,...,vm according to their order. We obtain a finite triangulation G∗. Apply Theorem 3.5 to obtain a circle packing P = {Cv}v∈V (G∗). By translating and dilating we may assume that Cv<sup>∗</sup> is centered at the origin and has radius 1. Apply the map <sup>z</sup> → <sup>1</sup> <sup>z</sup> on this packing. Since this map preserves circles, the image of the circles {Cv}v∈V (G∗)\{v∗} under this map is precisely the desired circle packing. 

Furthermore, we will require an auxiliary general estimate. Given a circle packing P and a set of vertices A, we write diam<sup>P</sup> (A) for the Euclidean diameter of the union of all circles in P corresponding to the vertices of A.

**Lemma 4.11** *Let* P *be a circle packing contained in* U *of a finite triangulation with boundary with maximum degree* Δ*, such that the circle of a chosen non-boundary vertex* v<sup>0</sup> *is centered at the origin and has radius* r0*. Assume that* r<sup>0</sup> ≥ rmin *for some constant* rmin > 0*. Then there exists a constant* c = c(rmin, Δ) > 0 *such that for any connected set* A *of vertices,*

$$\mathcal{R}\_{\rm eff}(\upsilon\_0 \leftrightarrow A) \ge c \log \frac{1}{\text{diam}\_P(A)}.\tag{4.1}$$

*If in addition all circles of the outer face are tangent to* ∂U *and* A *contains a vertex of the outer face, then*

$$\mathcal{R}\_{\text{eff}}(v\_0 \leftrightarrow A) \le c^{-1} \log \frac{1}{\text{diam}\_P(A) \land \frac{1}{2}} \,. \tag{4.2}$$

*Proof* Write ε = diam<sup>P</sup> (A) and let z(A) denote the union of all circles corresponding to the vertices of A. We begin with the proof of (4.1), which goes along similar lines to the proof of Lemma 4.9. Let <sup>z</sup><sup>0</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup> be such that z(A) ⊂ {|<sup>z</sup> <sup>−</sup> <sup>z</sup>0| ≤ <sup>ε</sup>}. For any r > 0 denote by Vr the set of vertices whose corresponding circles have centers inside {|z − z0| ≤ r}, so that A ⊂ Vε. Repeating the proof of Lemma 4.9 shows that there exists a constant C = C(Δ) > 0 such that


Regarding this proof, we note that it is possible that the set {|z − z0| ≤ r} is not contained in U (unlike the proof of Lemma 4.9 when the carrier is all of R2), however, this only works in our favor. The proof of (4.1) now proceeds similarly to the proof of Theorem 4.4 (3). When ε is small enough (depending only on rmin and Δ), by the Ring Lemma (Lemma 4.2), the Euclidean distance between the circle corresponding to v<sup>0</sup> and A is at least some constant (which again depends only on rmin and Δ) so that v<sup>0</sup> ∈ VCKε for some K = Ω(log(1/ε)). For each k = 0, 2, 4,...,K the sets of edges which have at least one endpoint in the annulus VCk+1<sup>ε</sup> \ VCkε are disjoint by (i). By (ii), each of these sets of edges contribute at least C−<sup>1</sup> to the energy of the unit current flow from A to v0, concluding the proof of (4.1) using Thomson's principle (Theorem 2.28).

For the proof of (4.2) we construct a unit flow from v<sup>0</sup> to A that has energy O(log(1/ε)). The construction is in the same spirit as the proof of Theorem 4.4 (4), but there are some technical difficulties to overcome. Since A contains a vertex that is tangent to <sup>∂</sup>U, we choose <sup>z</sup><sup>0</sup> <sup>∈</sup> <sup>∂</sup><sup>U</sup> that belongs to a circle of <sup>A</sup>. By rotating the packing we may assume that <sup>z</sup><sup>0</sup> <sup>=</sup> <sup>e</sup>iε/4.

We now treat two cases separately. In the first case we assume that there exists z<sup>1</sup> in z(A) such that arg(z1) ∈ [0, ε/2] and |z1| ≤ 1 − ε/2 such that the path in z(A) from z<sup>0</sup> to z<sup>1</sup> remains in the sector arg(z) ∈ [0, ε/2]. Consider the points

$$\mathbf{x}\_0 = -r\_0 \qquad \mathbf{x}\_1 = r\_0 \qquad \mathbf{y}\_1 = 1 - \varepsilon/3 \qquad \mathbf{y}\_0 = 1\,,$$

and note that x0, x<sup>1</sup> are the leftmost and rightmost points on the circle of v0. Let C<sup>0</sup> and C<sup>1</sup> be the upper half plane semi-circles in which x0, y<sup>0</sup> and x1, y<sup>1</sup> are antipodal points, respectively. The choice of y0, y<sup>1</sup> is made so that the path between z<sup>0</sup> to z<sup>1</sup> in z(A) must cross the region bounded by C0, C<sup>1</sup> and the intervals [x0, x1],[y1, y0], by our assumption on z<sup>1</sup> as long as ε is small enough. See Fig. 4.4, left.

For each t ∈ [0, 1] write Ct for the upper half plane semi-circle in which ty<sup>1</sup> + (1−t)y<sup>0</sup> and tx1+(1−t)x<sup>0</sup> are antipodal points, so that Ct continuously interpolates between C<sup>0</sup> and C1. See Fig. 4.4, left. Choose t ∈ [0, 1] uniformly at random and consider the random path γ which traces Ct from left to right. This random path

**Fig. 4.4** Left: for any t ∈ [0, 1] the semi-circle Ct must intersect the path in A between z<sup>0</sup> and z1. Right: the quadrilateral Qi is bounded between θi , θi+<sup>1</sup> , C<sup>0</sup> and C<sup>1</sup>

starts at the circle of v<sup>0</sup> and must hit the path between z<sup>0</sup> and z<sup>1</sup> by our previous discussion. Hence, the circles of P that intersect γ must contain a path in the graph from v<sup>0</sup> to A. By Claim 2.46 we obtain a flow I from v<sup>0</sup> to A whose energy *E*(I ) we now bound.

For an angle θ ∈ [0, π] we denote by wθ (t) the point at angle θ, seen from the center of Ct , on the semi-circle Ct . It is an exercise to see that the set of points {wθ (t) : t ∈ [0, 1]} form a straight line interval θ . Furthermore, when t is chosen uniform in [0, 1], the intersection of Ct and θ is a uniformly chosen point on θ . Set <sup>θ</sup><sup>0</sup> <sup>=</sup> 0 and θi <sup>=</sup> <sup>2</sup>i−1<sup>ε</sup> for <sup>i</sup> <sup>=</sup> <sup>1</sup>,...,K <sup>−</sup> 1 where <sup>K</sup> <sup>=</sup> O(log(1/ε)) such that θK−<sup>1</sup> ∈ [π/4,π/2] and set θK = π. We will obtain the bound *E*(I ) = O(K) by bounding from above by a constant the contribution to *E*(I ) coming from edges which intersect the quadrilateral Qi of <sup>R</sup><sup>2</sup> bounded by θi, θi+<sup>1</sup> , C0, C1; see Fig. 4.4, right. The random path γ restricted to Qi can be sampled by choosing a uniform random point on θi , setting t ∈ [0, 1] to be the unique number such that Ct intersects θi at the chosen point, and tracing the part of Ct from θi to θi+<sup>1</sup> . The lengths of the four curves bounding Qi are all of order 2<sup>i</sup> ε and so we deduce that if v corresponds to a circle of radius O(2<sup>i</sup> ε) which intersects Qi, then the probability that it is visited by γ is O(rad(v)/2<sup>i</sup> ε). Since the sum of rad(v)<sup>2</sup> over such v's is at most the area of Qi up to a multiplicative constant (note that some of these circles need not be contained in Qi) it is at most O(22<sup>i</sup> ε2). Since the degrees are bounded we deduce that the contribution to the energy from edges touching such v's is O(1). Lastly, if v corresponds to a larger circle, then we bound its probability of being visited by γ by 1 and note that there can only be O(1) many such v's whose circles intersects Qi. Thus the contribution from these is another O(1). Since there are O(log(1/ε)) such i's we learn that *E*(I ) = O(log(1/ε)) finishing our proof in this case using Thomson's principle (Theorem 2.28).

In the second case, we assume that there exists z<sup>1</sup> ∈ z(A) such that arg(z1) ∈ [0, ε/2] and |z1| ≥ 1 − ε. It is clear that since diam<sup>P</sup> (A) = ε either the first or the second case must occur. Denote z <sup>0</sup> = |z1|eiε/<sup>4</sup> and let <sup>x</sup>0, x<sup>1</sup> be antipodal points on the circle of v<sup>0</sup> such that the straight line between x<sup>0</sup> and x<sup>1</sup> is parallel to the straight line between z <sup>0</sup> and z1. The vertices z <sup>0</sup>, z1, x0, x<sup>1</sup> form a trapezoid, see Fig. 4.5. We choose a uniform random point t ∈ [0, 1] and stretch a straight line from tx<sup>0</sup> + (1 − t)x<sup>1</sup> to tz<sup>0</sup> + (1 − t)z <sup>1</sup>. We then continue it by a straight line from tz0+(1−t)z <sup>1</sup> to <sup>w</sup> <sup>∈</sup> <sup>∂</sup><sup>U</sup> where arg(w) <sup>=</sup> arg(tz<sup>0</sup> <sup>+</sup>(1−t)z <sup>1</sup>). Denote the resulting path by γt and note that it starts inside the circle of v<sup>0</sup> and must hit the path between z<sup>0</sup> and z<sup>1</sup> in z(A). Thus, the set of all circles which intersect γt form a path in the graph that starts at v<sup>0</sup> and ends at A; this random choice of γt gives us as usual a unit flow from v<sup>0</sup> to A by Claim 2.46. By repeating the same argument as in the previous case (that is, splitting the trapezoid into O(log(1/ε)) many trapezoids of constant aspect ratio), we see that the contribution to the energy of the flow induced by the random path γt of the edges in the trapezoid is O(log(1/ε)). Furthermore, the same argument gives that the edges in the quadrilateral formed by the vertices z0, z <sup>0</sup>, z<sup>1</sup> and e<sup>i</sup> arg(z1) also contribute at most a constant to the energy, concluding our proof for the second case by Thomson's principle again (Theorem 2.28).  **Fig. 4.5** The resistance across the trapezoid on vertices x0, x1, z <sup>0</sup>, z<sup>1</sup> is O(log(1/ε)) when |z <sup>0</sup> − z1| = Θ(ε)

*Proof of Theorem 4.4 (2)* Denote by dG(u, v) the graph distance between the vertices u, v of G. Fix some v<sup>0</sup> ∈ V and let

$$\begin{aligned} \mathcal{B}\_j &= \{ v : d\_G(v\_0, v) \le j \}, \\ V\_j &= \mathcal{B}\_j \cup \{ \text{finite components of } V \nmid \mathcal{B}\_j \}, \\ E\_j &= \{ \text{edges induced by } V\_j \}. \end{aligned}$$

The graph Gj = (Vj , Ej ) with the map structure inherited from G is a finite triangulation with boundary. Indeed, it is straightforward to check that it is 2 connected (i.e., the removal of a single vertex does not disconnect the graph) which implies that the outer face forms a simple cycle, see [20, Proposition 4.2.5]. Furthermore, since G is one-ended and we have added all the finite components in V \Bj there cannot be a face with more than 3 edges except for the outer face which we denote by ∂Gj .

Thus Gj is an increasing sequence of finite triangulations with boundary such that <sup>∪</sup><sup>j</sup> Gj <sup>=</sup> <sup>G</sup>. We apply Claim 4.10 to pack Gj inside the unit disk <sup>U</sup> such that the circles of ∂Gj are tangent to ∂U. By applying a Möbius transformation from U onto U, we may assume that the circle corresponding to v<sup>0</sup> is centered at the origin. We denote this packing by Pj and let r j <sup>0</sup> be the radius of v<sup>0</sup> in Pj .

Since G is transient it follows that there exists some c = c(Δ) > 0 such that r j <sup>0</sup> ≥ c for all j by Corollary 2.39. Indeed, if r j <sup>0</sup> ≤ ε, we learn by Lemma 4.9 and the proof of Theorem 4.4 (3) that *<sup>R</sup>*eff(v<sup>0</sup> ↔ ∞) <sup>≥</sup> <sup>c</sup> log(ε−1) for some c = c (Δ) > 0.

As we did in Claim 4.3, we now take a subsequence in which the centers and radii of all vertices converge. Denote the resulting limiting packing by P∞. This packing has all circles inside <sup>U</sup> and we therefore deduce that Carrier(P∞) <sup>⊆</sup> <sup>U</sup>. It is a priori possible that Carrier(P∞) is some strict subset of <sup>U</sup>, i.e., that all the circles stabilize inside some strict subset of U. We now argue that this is not possible.

Let Z be the set of accumulation points of Carrier(P∞); it suffices to show that <sup>Z</sup> <sup>⊂</sup> <sup>∂</sup><sup>U</sup> since any simply connected domain <sup>Ω</sup> <sup>⊂</sup> <sup>U</sup> for which ∂Ω <sup>⊂</sup> <sup>∂</sup><sup>U</sup> must equal <sup>U</sup>. Since <sup>Z</sup> is a compact set, let <sup>z</sup> <sup>∈</sup> <sup>Z</sup> minimize <sup>|</sup>z<sup>|</sup> among all <sup>z</sup> <sup>∈</sup> <sup>Z</sup>; it

suffices to show that <sup>z</sup> <sup>∈</sup> <sup>∂</sup>U. Fix ε > 0 and put

$$U\_{\varepsilon}(z) = \left\{ v \in G : |\mathsf{cent}\_{P\_{\infty}}(v) - z| \le \varepsilon \right\}.$$

The graph spanned on the vertices Uε(z) may be disconnected, yet by our choice of z it is clear that Uε(z) contains an infinite connected component. Indeed, one can draw a straight line from the origin to z without intersecting Z and consider the set of all circles intersecting this line; from some point onwards the vertices corresponding to these circles will reside in Uε(z).

Therefore, let Wε(z) be an infinite connected component of the graph spanned on Uε(z). Let J = J (z, ε) be the first integer such that VJ ∩ Wε(z) = ∅. Since the Gj 's are increasing finite sets and Wε(z) is an infinite connected set, we have that ∂Gj ∩ Wε(z) = ∅ for all j ≥ J . Consider now any connected component Aj of the graph spanned on the vertices Vj ∩ Wε(z).

Denote by P<sup>j</sup> <sup>∞</sup> the finite circle packing obtained from P<sup>∞</sup> by taking only the circles of Vj . It has the same adjacenty graph as Pj but it is a different packing. Since Aj ⊂ Wε(z), it follows that diamP<sup>j</sup> ∞(Aj ) <sup>≤</sup> <sup>4</sup>ε. By Lemma 4.11, Eq. (4.1), applied to the set Aj in the packing P<sup>j</sup> <sup>∞</sup>, we deduce that *R*eff(v<sup>0</sup> ↔ Aj ; Gj ) ≥ c log(1/ε). Since Aj is a connected component of Vj ∩ Wε(z) and since Wε(z) is an infinite connected set of vertices in G, it follows that Aj must contain a vertex of ∂Vj . Thus, we may apply Lemma 4.11, Eq. 4.2, to the set Aj , this time in the packing Pj , to get that there exists some c > 0 such that

$$\text{diam}\_{P\_{\mathcal{I}}}(A\_{\mathcal{I}}) \le \varepsilon^c. \tag{4.3}$$

Choose some vJ ∈ ∂GJ ∩ Wε(z) so that | centP∞(vJ ) − z| ≤ ε. For each j ≥ J choose vj ∈ ∂Gj ∩ Wε(z) so that vj and vJ are in the same connected component Aj of the graph spanned on Vj <sup>∩</sup> Wε(z). Since the circle of vj in Pj touches <sup>∂</sup><sup>U</sup> we learn by (4.3) that the distance of the circle of vJ in Pj from ∂U is at most ε<sup>c</sup> for all j ≥ J . Since the circle corresponding to vJ in P<sup>∞</sup> is the limit of its circles in Pj we deduce that the distance of centP∞(vJ ) from <sup>∂</sup><sup>U</sup> is at most <sup>ε</sup>c. Hence the distance of <sup>z</sup> from <sup>∂</sup><sup>U</sup> is at most <sup>ε</sup>+εc. Since <sup>ε</sup> was arbitrary we obtain that <sup>z</sup> <sup>∈</sup> <sup>∂</sup>U, as required. 

#### **4.5 Exercises**


3.(\*) Let P be a circle packing of a finite simple planar map with degree bounded by D such that all of its faces are triangles except for the outerface. Assume that the carrier of P is contained in [−11, 11] 2, contains [−10, <sup>10</sup>] <sup>2</sup> and that all circles have radius at most 1. Let h be the harmonic function taking the value 1 on all vertices with centers left of the line {−10} × <sup>R</sup>, taking the value 0 on all vertices with centers right of the line {10} × <sup>R</sup>, and is harmonic anywhere else. Assume x and y are two vertices such that their centers are contained in [−1, 1] <sup>2</sup> and that the Euclidean distance between these centers is at most > 0. Show that

$$|h(\mathbf{x}) - h(\mathbf{y})| \le \frac{C}{\log(1/\epsilon)},$$

for some constant C = C(D) > 0 independent of . [Hint: assume h(x) < h(y) and consider the sets A = {v : h(v) ≤ h(x)} and B = {v : h(v) ≥ h(y)}].

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# **Chapter 5 Planar Local Graph Limits**

#### **5.1 Local Convergence of Graphs and Maps**

In order to study large random graphs it is mathematically natural and appealing to introduce an infinite limiting object and study its properties. In their seminal paper, Benjamini and Schramm [11] introduced the notion of locally convergent graph sequences, which we now describe.

We will consider random variables taking values in the space *G*• of locally finite connected rooted graphs viewed up to root preserving graph isomorphisms. That is, *G*• is the space of pairs (G, ρ) where G is a locally finite graph (which may be finite or infinite) and ρ ∈ V (G) is a vertex of G and two elements (G1, ρ1), (G2, ρ2) are considered equivalent if there is a graph isomorphism between them (that is, a bijection ϕ : V (G1) → V (G2) such that ϕ(ρ1) = ϕ(ρ2) and {v1, v2} ∈ E(G1) if and only if {ϕ(v1), ϕ(v2)} ∈ E(G2)). We remark that throughout this book our graphs will almost entirely be connected. In the rare case when G is not connected, we impose the convention that (G, ρ) refers to (G[ρ],ρ) where G[ρ] is the connected component of ρ in G. This way (G, ρ) ∈ *G*• even when G is disconnected (this will only be relevant in Chap. 6, and in particular in Lemma 6.11 and its usage).

In a similar fashion we define *M*• to be the set of equivalence classes of rooted maps; in this case we require the graph isomorphism to preserve additionally the cyclic permutations of the neighbors of each vertex, that is, it is a **map isomorphism**. Let us describe the topology on *G*• and *M*•. For convenience we discuss *G*• but every statement in the following holds for *M*• as well.

Given an element (G, ρ) of *G*•, the finite graph BG(ρ, R) is the subgraph of (G, ρ) rooted at ρ spanned by the vertices of distance at most R from ρ. We provide *G*• with a metric dloc

$$d\_{\mathrm{loc}}((G\_1, \rho\_1), (G\_2, \rho\_2)) = 2^{-R},$$

© The Author(s) 2020

A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes in Mathematics 2243, https://doi.org/10.1007/978-3-030-27968-4\_5

where R is the largest integer for which BG<sup>1</sup> (ρ1, R) and BG<sup>2</sup> (ρ2, R) are isomorphic as graphs. This is a separable topological space (the finite graphs form a countable base for the topology) and is easily seen to be complete, thus it is a Polish space. The distances are bounded by 1 but the space is not compact. Indeed, the sequence Gn of stars with n leaves emanating from the root ρ has no converging subsequence.

Since *G*• is a Polish space, we can discuss convergence in distribution of a sequence of random variables {Xn} ∞ <sup>n</sup>=<sup>1</sup> taking values in *<sup>G</sup>*•. We say that Xn

**converges in distribution** to a random variable X, and denote it by Xn <sup>d</sup> −→ <sup>X</sup>, if for every bounded continuous function <sup>f</sup> : *<sup>G</sup>*• <sup>→</sup> <sup>R</sup> we have that <sup>E</sup>(f (Xn)) <sup>→</sup> E(f (X)). We will be focused here on the particular situation in which Xn is a *finite* rooted random graph (Gn, ρn) such that given Gn, the root ρn is uniformly distributed among the vertices of Gn. It is a very common setting and justifies the following definition.

**Definition 5.1** Let {Gn} be a sequence of (possibly random) finite graphs. We say that Gn **converges locally** to a (possibly infinite) random rooted graph (U, ρ) ∈ *G*•, and denote it by Gn loc −→ (U, ρ), if for every integer r ≥ 1,

$$B\_{G\_n}(\rho\_n, r) \stackrel{d}{\longrightarrow} B\_U(\rho, r),$$

where ρn is a uniformly chosen vertex from Gn.

It is straightforward to see that this definition is equivalent to saying that the random variables (Gn, ρn) converge in distribution to (U, ρ). Note that this definition is consistent whether Gn is a deterministic finite graph or is a random variable drawn from some probability measure. In both cases BGn (ρn,r) is a random variable taking values in *G*• and we clarify that ρn is drawn uniformly *conditioned* on Gn.

#### *Examples*



Our goal in this chapter is to prove the following pioneering result.

**Theorem 5.2 (Benjamini–Schramm [11])** *Let* M < ∞ *and let* Gn *be finite planar maps (possibly random) with degrees almost surely bounded by* M *such that* Gn loc −→ (U, ρ)*. Then* (U, ρ) *is almost surely recurrent.*

For instance, a local limit of planar maps cannot be the 3-regular infinite tree (however, the 3-regular infinite tree can be obtained as a local limit of uniformly random 3-regular graphs). The bounded degree assumption in Theorem 5.2 is necessary. Indeed, suppose we start with a binary tree of height n and replace each edge (u, v) that is at distance <sup>k</sup> <sup>≥</sup> 0 from the leaves by 2<sup>k</sup> parallel edges. By the same reasoning of the local convergence of binary trees to the canopy tree, the modified graph sequence converges locally to a modified canopy tree in which an edge at distance k from the leaves is replaced with 2<sup>k</sup> parallel edges. Using the parallel law it is immediate to see that this graph is transient, and that the effective resistance from a leaf to ∞ is at most 2 (in fact it equals 2). See Fig. 5.2.

#### **5.2 The Magic Lemma**

Suppose <sup>C</sup> <sup>⊆</sup> <sup>R</sup><sup>2</sup> is finite. For each <sup>w</sup> <sup>∈</sup> <sup>C</sup>, define

$$\rho\_w = \min\{|v - w| \, : \, v \in C \backslash \{w\}\}.$$

We call ρw the **isolation radius** of w. Given δ ∈ (0, 1), s ≥ 2 and w ∈ C, we say that w is (δ, s)**-supported** if in the disk of radius δ−<sup>1</sup>ρw around w there are at least s points of C outside any given disk of radius δρw. In other words, w is (δ, s)**-supported** if

$$\inf\_{p \in \mathbb{R}^2} \left| C \cap B\left(w, \delta^{-1} \rho\_w\right) \backslash B(p, \delta \rho\_w) \right| \ge s. \varepsilon$$

The proof of Theorem 5.2 is based on the following lemma, which has been dubbed "the Magic Lemma".

**Lemma 5.3 ([11])** *There exists* A > 0 *such that for every* δ ∈ (0, 1/2)*, every finite* <sup>C</sup> <sup>⊆</sup> <sup>R</sup><sup>2</sup> *and every* <sup>s</sup> <sup>≥</sup> <sup>2</sup>*, the number of* (δ, s)*-supported points in* <sup>C</sup> *is at most*

$$\frac{A|C|\delta^{-2}\ln(\delta^{-1})}{s}.$$

*Remark 5.4* We prove the lemma for R2, but it holds for R<sup>d</sup> or any other doubling metric space. In fact, a metric space for which the lemma holds must be doubling; see [29].

#### *Proof of Lemma 5.3*

Let <sup>k</sup> <sup>≥</sup> 3 be an integer (later we will take <sup>k</sup> <sup>=</sup> k(δ)). Let <sup>G</sup><sup>0</sup> be a tiling of <sup>R</sup><sup>2</sup> by <sup>1</sup> <sup>×</sup> 1 squares, rooted at some point <sup>p</sup>, and for every <sup>n</sup> <sup>∈</sup> <sup>Z</sup>, let Gn be a tiling of <sup>R</sup><sup>2</sup> by <sup>k</sup><sup>n</sup> <sup>×</sup> <sup>k</sup><sup>n</sup> such that each square of Gn is tiled by <sup>k</sup> <sup>×</sup> <sup>k</sup> squares of Gn−1. We may choose p so that none of the points of C lies on the edge of a square.

We say that a square S ∈ Gn is s**-supported** if for every smaller square S ∈ Gn−<sup>1</sup> we have that |C ∩ (S \ S )| ≥ s.

*Claim 5.5* For any s ≥ 2 the total number of s-supported squares, in G = <sup>n</sup>∈<sup>Z</sup> Gn, is at most 2|C|/s.

*Proof* Define a "flow" <sup>f</sup> : <sup>G</sup> <sup>×</sup> <sup>G</sup> <sup>→</sup> <sup>R</sup> as follows:

$$f(S',S) = \begin{cases} \min(s/2, |S' \cap C|) & S' \subseteq S, S' \in G\_n, S \in G\_{n+1}, \\ -f(S, S') & S \subseteq S', S \in G\_n, S' \in G\_{n+1}, \\ 0 & \text{otherwise}. \end{cases}$$

Let us make two initial observations. First we have that

$$\sum\_{S' \in G} f(S', S) \ge 0 \,, \tag{5.1}$$

by splitting into the two cases depending on whether there exists a square S ⊆ S such that f (S , S) = s/2 or not. Secondly, if S is a s-supported square

$$\sum\_{S' \in G} f(S', S) \ge \frac{s}{2},\tag{5.2}$$

by splitting into cases depending on whether the number of squares S ⊆ S such that f (S , S) = s/2 is at most one or at least two.

Let <sup>a</sup> <sup>∈</sup> <sup>Z</sup> be such that each square in Ga contains at most 1 point of <sup>C</sup> so there are no s-supported squares in <sup>n</sup>≤<sup>a</sup> Gn. It easily follows from the definition of <sup>f</sup> that

$$\sum\_{S' \in G\_d} \sum\_{S \in G\_{d+1}} f(S', S) = |C|,\tag{5.3}$$

and that for every <sup>b</sup> <sup>∈</sup> <sup>Z</sup>

$$\sum\_{S' \in G\_b} \sum\_{S \in G\_{b+1}} f(S', S) \ge 0. \tag{5.4}$$

Now, using (5.3) and (5.4),

$$\sum\_{n=a+1}^{b} \sum\_{S \in G\_n} \sum\_{S' \in G} f(S', S) = \sum\_{n=a+1}^{b} \sum\_{S \in G\_n} \left( \sum\_{S' \in G\_{n-1}} f(S', S) + \sum\_{S' \in G\_{n+1}} f(S', S) \right)$$

$$= \sum\_{S \in G\_{a+1}} \sum\_{S' \in G\_{b}} f(S', S) + \sum\_{S \in G\_b} \sum\_{S' \in G\_{b+1}} f(S', S) \le |C|.$$

Therefore, using ( 5.1) and (5.2), we deduce that there are at most 2|C|/s squares in <sup>b</sup>≥n>a Gn that are <sup>s</sup>-supported. Sending <sup>b</sup> → ∞ finishes the proof. 

The above claim is very close to the statement of Lemma 5.3 which we are pursuing. However, we need to move from squares to circles. We choose k = <sup>20</sup>δ−<sup>2</sup> and let <sup>β</sup> <sup>∼</sup> Unif([0,ln <sup>k</sup>]). Let <sup>G</sup><sup>0</sup> be a tiling with side length <sup>e</sup>β, based at the origin. Suppose we have defined Gn as a tiling of squares of side length eβkn; then Gn+<sup>1</sup> is a tiling of squares of side length <sup>e</sup>βkn+<sup>1</sup> that is based uniformly at one of the k<sup>2</sup> possible points of Gn. Because the desired statement is invariant under translation and dilation of C, we may assume that C does not intersect the edges of Gn (for every n) and that ρw ≥ k for every w ∈ C. We call a point w ∈ C a **city** in a square S ∈ G if:


*Claim 5.6* The probability that any given <sup>w</sup> <sup>∈</sup> <sup>C</sup> is a city is Ω(ln−1(δ−1)).

*Proof* For the first item to hold, <sup>β</sup> needs to satisfy that there exists <sup>n</sup> <sup>∈</sup> <sup>Z</sup> such that <sup>e</sup>βkn ∈ [4δ−<sup>1</sup>ρw, <sup>5</sup>δ−<sup>1</sup>ρw], or <sup>β</sup> <sup>+</sup> <sup>n</sup> ln <sup>k</sup> <sup>∈</sup> ln(δ−<sup>1</sup>ρw) + [ln 4,ln 5]. Since <sup>β</sup> <sup>∈</sup> Unif([0,ln <sup>k</sup>]), the probability for that is (ln(5/4))/ ln <sup>k</sup>, which is Ω(ln−1(δ−1)) when δ ∈ (0, 1/2).

As for the second item, it holds with positive probability (independent of δ) over the <sup>k</sup><sup>2</sup> choices for basing Gn on top of Gn−1, given that <sup>β</sup> satisfies the requirement posed by the first item. 

*Claim 5.7* If w is a city in S and is (δ, s)-supported, then S is s-supported.

*Proof* If S ∈ Gn is as above, then any little square S ∈ Gn−<sup>1</sup> has side length at most

$$\frac{\delta^2}{20} \cdot \frac{\mathsf{S}\rho\_w}{\delta} = \frac{\delta \rho\_w}{4}.$$

Hence, it is contained in a disk of radius δρw. Thus, for every S ∈ Gn−<sup>1</sup> with S ⊆ S there exists a point p such that

$$|\mathcal{C}\cap(\mathcal{S}\backslash\mathcal{S}')| \ge \left|\mathcal{C}\cap\left(\mathcal{B}\left(w,\delta^{-1}\rho\_w\right)\backslash\mathcal{B}(p,\delta\rho\_w)\right)\right| \ge s\,,,$$

where we have used the fact that B w, δ−<sup>1</sup>ρw ⊂ S. 

Now note that the expected number of pairs (w, S) such that S is s-supported, w is (δ, s)-supported, and w is a city, is at least c ln−1(δ−1)N, where N is the number of (δ, s)-supported points. Also, no more than cδ−<sup>2</sup> points of C can be cities in a single square S. It follows from Claim 5.5 that

$$N \le \frac{A|C|\delta^{-2}\ln(\delta^{-1})}{s},$$

concluding the proof of Lemma 5.3. 

#### **5.3 Recurrence of Bounded Degree Planar Graph Limits**

Theorem 5.2 follows immediately from the following theorem which gives a quantitative estimate on the growth of the resistance in local limits of bounded degree planar maps. In particular, it states that the resistance grows logarithmically in the Euclidean distance of the corresponding circle packing.

**Theorem 5.8** *Let* (U, ρ) *be a local limit of (possibly random) finite planar maps with maximum degree at most* D*. Then, almost surely, there exist a constant* c > 0 *and a sequence* {Bk }k≥<sup>1</sup> *of subsets of* U *such that for each* k *we have*

*1.* <sup>|</sup>Bk <sup>|</sup> <sup>≤</sup> <sup>c</sup>−1k*, and 2. R*eff(ρ ↔ U \ Bk) ≥ c log k*.*

*In particular,* (U, ρ) *is almost surely recurrent.*

We write <sup>B</sup>euc(p, r) for the Euclidean ball of radius <sup>r</sup> around a point <sup>p</sup> <sup>∈</sup> <sup>R</sup>2. As before, for a subset <sup>O</sup> <sup>⊂</sup> <sup>R</sup><sup>2</sup> and a given circle packing we write VO for the set of vertices in which the centers of the corresponding circles are in O. In order to prove Theorem 5.8, we will need the following immediate corollary of the Magic Lemma (Lemma 5.3):

**Corollary 5.9** *Let* G *be a finite simple planar triangulation, and* P *a circle packing of* G*. Let* ρ *be a uniform random vertex and* P *a dilation and translation of* P *such that the circle of* ρ *is a unit circle centered at the origin* **0***. Then, there exists a universal constant* A > 0 *such that in the packing* P *, for every real* r ≥ 2 *and integer* s ≥ 2

$$\mathbb{P}\left(\forall p \in \mathbb{R}^2 \quad \left| V\_{B\_{\text{euc}}(\mathbf{0}, r) \backslash B\_{\text{euc}}(p, \frac{1}{r})} \right| \geq s \right) \leq \frac{Ar^2 \log r}{s}.$$

*Proof* Apply the Magic Lemma with <sup>δ</sup> <sup>=</sup> <sup>1</sup> <sup>r</sup> and s = s, with the centers of circles of P as the point set C. Note that there exists a constant C > 0 such that for all

w ∈ V the isolation radius of w, ρw, satisfies rad(Cw) ≤ ρw ≤ C rad(Cw) (without appealing to the Ring Lemma). 

The following lemma provides the main estimate needed to prove Theorem 5.8. Once it has been shown, Theorem 5.8 will follow by a Borel-Cantelli argument.

**Lemma 5.10** *Let* G *be a finite simple planar map with maximum degree at most* D *and let* ρ *be a uniform random vertex of* G*. Then, there exists a constant* C = C(D) < ∞ *such that for all* k ≥ 1*,*

$$\mathbb{P}\left(\exists B \subseteq V, \ |B| \le Ck, \ \mathcal{R}\_{\text{eff}}(\rho \leftrightarrow V \backslash B) \ge C^{-1}\log k\right) \ge 1 - Ck^{-\frac{1}{3}}\log k,$$

*where we interpret R*eff(ρ ↔ V \ B) = ∞ *when* B = V *.*

*Proof* We first assume that G is a triangulation and consider a circle packing of it where the circle of ρ is a unit circle centered at the origin **0**. Applying Corollary 5.9 with r = k 1 <sup>3</sup> , s <sup>=</sup> <sup>k</sup>, we have that with probability at least 1 <sup>−</sup> Ak−<sup>1</sup> <sup>3</sup> log(k)/3, there exists <sup>p</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup> with

$$\left| V\_{B\_{\mathrm{euc}}(\mathbf{0},r)\backslash B\_{\mathrm{euc}}(p,\frac{1}{r})} \right| < k \,.$$

Now, if |VBeuc(p, <sup>1</sup> r ) | ≤ 1, we set B = VBeuc(**0**,r). We then have |B| ≤ k and by applying Ω(log k) times Lemma 4.9 together with the series law (Claim 2.24) we get that *R*eff(ρ ↔ V \B) ≥ c log k for some c = c(D) > 0. Else, if |VBeuc(p, <sup>1</sup> r ) | ≥ 2 then we take B = VBeuc(**0**,r) \ VBeuc(p, <sup>1</sup> r ) . By the Ring Lemma, there exists a c = c (D) > 0 such that |p| ≥ 1+c . Since |VBeuc(p, <sup>1</sup> r ) | ≥ 2, we have a vertex in that set with radius at most r−1. Therefore, Beuc(p, <sup>2</sup> <sup>r</sup> ) contains at least one full circle Cv. Hence, by scaling and translating such that Cv <sup>=</sup> <sup>U</sup>, we get (again, by Lemma 4.9) that

$$\mathcal{R}\_{\text{eff}}\left(V\_{B\_{\text{euc}}(p,\frac{2}{r})} \leftrightarrow V \mid V\_{B\_{\text{euc}}(p,c'/2)}\right) \cong c2\log k\,,$$

for some other constant c<sup>2</sup> = c2(D) > 0. Since ρ ∈ V \ VBeuc(p,c /2) we obtain

$$\mathcal{R}\_{\text{eff}}\left(\rho \leftrightarrow V\_{B\_{\text{vac}}(p,\frac{2}{r})} \right) \cong c\_2 \log k\dots$$

By Lemma 4.9 we also have that

$$\mathcal{R}\_{\text{eff}}\left(\rho \leftrightarrow V \mid V\_{B\_{\text{cuc}}(\mathbf{0}, r)}\right) \cong c\_3 \log k\,,$$

for some c<sup>3</sup> = c3(D) > 0. By Claim 2.22 this means that

$$\mathbb{P}\_{\rho} \left( \mathsf{\tau}\_{V \backslash V\_{\mathsf{Buc}(\mathbf{0}, r)}} < \mathsf{\tau}\_{\rho}^{+} \right) \leq \frac{1}{c\_{2} \log(k)} \quad \text{and} \quad \mathbb{P}\_{\rho} \left( \mathsf{\tau}\_{V\_{\mathsf{Buc}(\rho, \frac{2}{r})}} < \mathsf{\tau}\_{\rho}^{+} \right) \leq \frac{1}{c\_{3} \log(k)}.$$

By the union bound

$$\mathbb{P}\_{\rho} \left( \mathfrak{r}\_{V \backslash B} < \mathfrak{r}\_{\rho}^{+} \right) \le \frac{2}{\min(c\_2, c\_3) \log(k)},$$

hence by Claim 2.22 again

$$\mathcal{R}\_{\text{eff}}\,(\rho \leftrightarrow V \backslash B) \ge \min(c\_2, c\_3) D^{-1} \log(k) / 2 \,,$$

concluding the proof when G is a triangulation.

If G is not a triangulation, we would like to add edges to make it a triangulation while making sure that the maximal degree does not increase too much. We also have to ensure that the graph remains simple which may require us to add some additional vertices as well. Let f be a face of G with vertices v1,...,vk . Suppose first that there are no edges between non-consecutive vertices of the face. In this case, we draw the edges in a zig-zag fashion, as in Fig. 5.3.

In the case where there are edges between non-consecutive vertices of the face exist, we draw a cycle u1,...,uk inside f . Then, we connect ui to vi and vi+<sup>1</sup> for each i<k and uk to vk and v1. Finally, we triangulate the inner face created by the new cycle by zig-zagging as in the previous case (see Fig. 5.4).

Since each vertex of the original graph is a member of at most D faces and for each face at most 2 edges are added, the maximal degree of the resulting graph is at most 3D. Similarly, the number of vertices in the resulting graph is at most D times the number of vertices in the original graph hence the probability of a random vertex being a vertex of the original graph is at least D−1. If this occurs then it is straightforward to see that the existence of a subset of vertices B in the new graph which satisfies the required conditions implies the existence of such a set in the old graph, concluding our reduction to the triangulation case and finishing our proof. 

We are ready to deduce Theorem 5.8.

*Proof of Theorem 5.8* Assume that Gn are finite planar maps with maximum degree at most D such that Gn loc −→ (U, ρ). If {Gn} are not simple graphs we erase selfloops and merge parallel edges into a single edge to obtain the sequence {G <sup>n</sup>}. It is immediate that G n loc −→ (U , ρ ) where (U , ρ ) is distributed as (U, ρ) after removing from U all loops and merging parallel edges into a single edge. Since the maximum degree is bounded, U is recurrent if and only if U is recurrent. Thus we may assume that Gn are simple graphs so the previous estimates may be used.

Denote by *A*<sup>k</sup> the event

$$\mathcal{A}\_k = \{ \exists B \subseteq U, \ |B| \le Ck, \ \mathcal{R}\_{\text{eff}} \ (\rho \leftrightarrow V \ \langle \ B \rangle \ge c \log k \} \ ,\ \}$$

where <sup>C</sup> <sup>=</sup> C(D) < <sup>∞</sup> is the constant from Lemma 5.10. Therefore <sup>P</sup>(*A*<sup>c</sup> <sup>k</sup>) ≤ c−1k−<sup>1</sup> <sup>3</sup> log(k). Looking at the sequence {*A*2<sup>j</sup> }j≥1, by Borel-Cantelli, almost surely there exists j<sup>0</sup> such that for all j ≥ j<sup>0</sup> the event *A*2<sup>j</sup> holds. Thus we have proved the required assertion for k which is a power of 2. To prove this for all k sufficiently large, let B2<sup>j</sup> be the set guaranteed to exist in the definition of *A*2<sup>j</sup> , and take Bk = <sup>B</sup>2<sup>j</sup> for the unique <sup>j</sup> for which 2<sup>j</sup> <sup>≤</sup> k < <sup>2</sup>j+1. It is immediate that these sets satisfy the assertion of the theorem, concluding our proof. 

#### **5.4 Exercises**


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# **Chapter 6 Recurrence of Random Planar Maps**

Our main goal in this chapter is to remove the bounded degrees assumption in Theorem 5.2 and replace it with the assumption that the degree of the root has an exponential tail.

**Theorem 6.1 ([31])** *Let* Gn *be a sequence of (possibly random) planar graphs such that* Gn loc −→ (U, ρ) *and there exist* C, c > <sup>0</sup> *such that* <sup>P</sup>(deg(ρ) <sup>≥</sup> k) <sup>≤</sup> Ce−ck *for every* k*. Then* U *is almost surely recurrent.*

As discussed in Sect. 1.2, the last theorem is immediately applicable in the setting of random planar maps. It is well known that the degree of the root in the UIPT and the UIPQ has an exponential tail. See [5, Lemma 4.1 and 4.2] or [26] for the UIPT and [8, Proposition 9] for the UIPQ.

**Corollary 6.2 ([31])** *The UIPT/UIPQ are almost surely recurrent.*

#### **6.1 Star-Tree Transform**

We present here a transformation which transforms any planar map G to a planar map G<sup>∗</sup> with maximal degree of 4. We call this transformation G → G<sup>∗</sup> the **startree transform**. Recall that a **balanced rooted tree** is a finite rooted tree in which every non-leaf vertex has precisely two children and the distance of the leaves from the root differs by at most 1. The transformation is performed as follows.


**Fig. 6.1** The star-tree transform. (**a**) An original edge of G. (**b**) Subdividing an edge. (**c**) The "star" of a vertex in G . (**d**) Transforming the star of v into a tree Tv

preserves planarity. Furthermore, add two extra vertices and attach them to the root. Denote this tree by Tv. See Fig. 6.1d.

*Remark 6.3* The careful reader will notice that we have not specified precisely what is Tv (if degG(v) is not a power of 2 there may several balanced binary trees with deg(v) leaves) and in which way precisely we identify the leaves of Tv with the neighbors of v in G (we may rotate the tree and get a different identification while still preserving planarity). This is a subtle yet important issue1 and our convention is that the choice of tree and identification are performed uniformly at random from all the possible choices. This will be crucially used in Claim 6.13.

**Lemma 6.4** *Let* G *be a planar map and* G∗ *its star-tree transform. We set edge resistances on* G<sup>∗</sup> *by putting* Re = 1/dG(v)*, where* v *is the vertex of* G *for which* e ∈ Tv *and* dG(v) *is the degree of* v *in* G*. If the network* (G∗, Re) *is recurrent, then* G *is recurrent as well.*

*Proof* It is clear that from the point of view of recurrence versus transience, the two edges leading to the two "extra" neighbors of each root do not matter and can be removed. Hence for the rest of the proof we write Tv for the previously defined tree with these two edges removed. The purpose of these extra edges will become apparent later in the proof of Theorem 6.1.

Assume G is transient and let a ∈ V (G) be some vertex. There is a flow θ from a to ∞ such that *E*(θ ) < ∞. We will construct a flow θ <sup>∗</sup> on (G∗, Re) from a to

<sup>1</sup>We thank Daniel Jerison for pointing this out to us.

∞ with finite energy, showing that (G∗, Re) is transient, giving the theorem. First we define a flow θ from a to infinity in G in the natural manner: for each edge e = (x, y) of G we set θ (x, we) = θ (we,y) = θ (x, y). Obviously *E*(θ ) = 2*E*(θ ).

Next we provide some notation. We denote by A the set of vertices that were added to form G in the first step of the star-tree transform, that is, the white vertices in Fig. 6.1. Each vertex w ∈ A is a leaf of precisely two trees Tu and Tv, where {u, v} was the edge of G that w divided. We call u and v the **tree roots** of w. We denote by B the set of vertices that were added to G∗ in the second step of the startree transform, that is, the gray vertices in Fig. 6.1d. The vertices of V (G) are the black discs in Fig. 6.1. Each vertex of x ∈ V (G) ∪ B is a member of a single tree Tv; we call v the **tree root** of x. Lastly, for any x ∈ V (G)∪B we denote by Cx ⊂ A the set of leaves of Tv, where v is the tree root of x, for which the path from the leaf to the root of Tv goes through x; in other words, Cx is the set of leaves of Tv which are the "descendants" of x. If x ∈ A, then we set Cx = {x}.

To define θ <sup>∗</sup>, let e = (x, y) be an edge of Tv. Assume that x is closer to the root of Tv than y in graph distance. We set

$$\theta^\*(e) = \sum\_{w \in C\_{\mathcal{Y}}} \theta'(v, w) \,.$$

The construction of θ ∗ is depicted in Fig. 6.2.

We will now show that *E*(θ <sup>∗</sup>) ≤ 2*E*(θ ) where the energy of θ ∗ is taken in the network (G∗, Re). Let v ∈ V (G) and write h for the height of Tv, that is, h is the maximal graph distance from a leaf of Tv to its root. Note that since the tree

**Fig. 6.2** The construction of the flow θ<sup>∗</sup> from θ. (**a**) An original edge of G which has flow θ1. (**b**) The flow passes through the divided edge. (**c**) The flow going out from a vertex of G in G . (**d**) The division of the flow in Tv

is balanced, the distances from the leaves to the root vary by at most 1. Let e = (x, y) be an edge of Tv and assume that x is closer than y to the root of Tv. By the construction of θ <sup>∗</sup>, the contribution of e to *E*(θ <sup>∗</sup>) is

$$\left(R\_{\epsilon}\theta^\*(e)\right)^2 = \frac{1}{d\_G(v)}\left(\sum\_{w \in C\_{\mathcal{Y}}} \theta'(v, w)\right)^2.$$

If the graph distance of <sup>y</sup> from the root is ∈ {1,...,h}, then <sup>|</sup>Cy| ≤ <sup>2</sup>h−. Hence by Cauchy-Schwarz

$$\mathcal{R}\_{\epsilon} \theta^\*(e)^2 \le \frac{2^{h-\ell}}{d\_G(v)} \sum\_{w \in C\_{\mathcal{Y}}} \theta'(v, w)^2 \dots$$

Summing over all edges in Tv at distance from the root, we go over each leaf of Tv precisely once. Thus,

$$\sum\_{\substack{e=(\chi,\chi)\in T\_v\\d\_{G^\ast}(\chi,v)=\ell}} R\_e \theta^\ast(e)^2 \le \frac{2^{h-\ell}}{d\_G(v)} \sum\_{w \in C\_v} \theta'(v,w)^2 \dots$$

We now sum over all edges in Tv by summing over ∈ {1,...,h}. We get

$$\sum\_{e \in T\_v} R\_e \theta^\*(e)^2 \le \frac{2^h}{d\_G(v)} \sum\_{w \in C\_v} \theta'(v, w)^2 \le 2 \sum\_{w \in C\_v} \theta'(v, w)^2 \ll 1$$

since h ≤ log2(dG(v)) + 1. Lastly, we sum this over all v ∈ V (G) to obtain that

$$
\mathcal{E}(\theta^\*) \le 2\mathcal{E}(\theta') = 4\mathcal{E}(\theta)\,,$$

concluding our proof. 

#### **6.2 Stationary Random Graphs and Markings**

#### *Stationary Random Graphs*

Recall that Theorem 6.1 and the entire setup of Chap. 5 is adapted to the case when Gn is itself random. The reason is that in Definition 5.1 we consider the graph distance ball BGn (ρn,r) as a random variable in the probability space (*G*•, dloc), where ρn conditioned on Gn is a uniformly chosen random vertex.

Let us emphasize that this is **not** the same as drawing a sample of {Gn} and claiming that almost surely Gn loc −→ (U, ρ). For example, let Gn be a path of length n with probability 1/2 and an n×n square grid with probability 1/2, independently for all n. In this case Gn loc −→ (U, ρ) where <sup>U</sup> <sup>=</sup> <sup>Z</sup> with probability 1/2 and <sup>U</sup> <sup>=</sup> <sup>Z</sup><sup>2</sup> with probability 1/2, however, almost surely on the sequence {Gn}, the local limit of Gn does not exist.

In many cases it is useful to take a random root drawn from the stationary distribution on Gn, that is, the probability distribution on vertices giving each vertex v probability degGn (v)/2|E(Gn)|. In a similar fashion to Definition 5.1, we define this type of local convergence.

**Definition 6.5** Let {Gn} be a sequence of (possibly random) finite graphs with nonempty sets of edges. We say that Gn loc −→<sup>π</sup> (U, ρ) where (U, ρ) is a random rooted graph, if for every integer r ≥ 1,

$$B\_{G\_n}(\rho\_n, r) \stackrel{d}{\longrightarrow} B\_U(\rho, r),$$

where ρn is a randomly chosen vertex from Gn with distribution proportional to the vertex degrees. We call such a limit a **stationary local limit**.

Let us remark that Gn loc −→ (U, ρ) does not imply that Gn loc −→<sup>π</sup> (U , ρ ) for some (U , ρ ). Indeed, let Gn be a path of length <sup>n</sup> attached to a complete graph on <sup>√</sup><sup>n</sup> vertices. Then the local limit of Gn is Z, however the limit according to a stationary random root does not exist.

The reason for taking the loc −→<sup>π</sup> limit rather than the uniform limit as before is that the random walk on the limit (U, ρ) starting from ρ is then stationary.

*Claim 6.6* Assume that Gn loc −→<sup>π</sup> (U, ρ). Conditioned on (U, ρ), let <sup>X</sup><sup>1</sup> be a uniformly chosen neighbor of ρ. Then (U, X1) is equal in law to (U, ρ). Similarly, if {Xn}n≥<sup>0</sup> is the simple random walk on (U, ρ), then for each n ≥ 0 the law of (U, Xn) coincides with the law of (U, ρ).

*Proof* If H is a finite graph and v is a vertex chosen with probability proportional to its degree, then it is immediate that a uniformly chosen random neighbor of v is distributed according to the stationary distribution. Thus for any fixed r > 0 the ball BGn (ρn,r) has the same distribution as BGn (X1,r) where ρn is drawn from the stationary distribution on Gn and X<sup>1</sup> is a uniform neighbor of ρn. The claim follows now by definition. 

**Definition 6.7** A random rooted graph (G, ρ) is called a **stationary random graph** if (G, X1) has the same distribution as (G, ρ), where the vertex X<sup>1</sup> is a uniform neighbor of ρ conditioned on (G, ρ).

We would like to develop a simple abstract framework that will allow us to comfortably move from loc −→ convergence to loc −→<sup>π</sup> convergence and vice versa. This is straightforward when {Gn} are a sequence of *deterministic* graphs with uniformly bounded average degree but is less obvious when Gn themselves are random. For this we need to *degree bias* our random graphs.

**Definition 6.8** Denote by P the law of a random rooted graph (G, ρ) and assume that <sup>E</sup>deg(ρ) <sup>∈</sup> (0,∞). The probability measure <sup>μ</sup> on (*G*•, dloc) defined by

$$\mu(\mathcal{A}) := \frac{1}{\mathbb{E}\,\mathrm{deg}(\rho)} \sum\_{k \ge 1} k \,\,\mathbb{P}(\mathcal{A} \cap \{\mathrm{deg}(\rho) = k\})\,\,.$$

for any event *<sup>A</sup>* <sup>⊂</sup> (*G*•, dloc) is called the **degree biasing** of <sup>P</sup>. Similarly, if we assume that almost surely ρ is not an isolated vertex, then the probability measure ν defined by

$$\nu(\mathcal{A}) = \frac{1}{\mathbb{E}[\deg(\rho)^{-1}]} \sum\_{k \ge 1} \frac{\mathbb{P}(\mathcal{A} \cap \{\deg(\rho) = k\})}{k},$$

is called the **degree unbiasing** of P.

**Lemma 6.9** *Assume that* (G, ρ) *is a random rooted graph such that* G *is almost surely finite, that the distribution of* <sup>ρ</sup> *given* <sup>G</sup> *is uniform and that* <sup>E</sup>deg(ρ) <sup>∈</sup> (0,∞)*. Then the degree biasing of* (G, ρ) *is a stationary random graph.*

*Conversely, assume that* (G<sup>π</sup> , ρ<sup>π</sup> ) *is a stationary random graph such that* G<sup>π</sup> *is almost surely finite and has no isolated vertices. Then its degree unbiasing* (G, ρ) *is such that* G *is almost surely finite and* ρ *conditioned on* G *is uniformly distributed.*

*Proof* We will prove only the first statement and the second is similar. Denote by (G<sup>π</sup> , ρ<sup>π</sup> ) a random variable drawn according to the degree biasing of (G, ρ). Let H be a fixed finite graph and denote by deg<sup>H</sup> (v) the degree of a vertex v in H. By definition we have that

$$\mathbb{P}((G^{\mathbb{Z}}, \rho^{\pi}) = (H, v)) = \frac{\deg\_H(v) \cdot \mathbb{P}((G, \rho) = (H, v))}{\mathbb{E}\, \text{deg}(\rho)} \,. \tag{6.1}$$

Let X<sup>1</sup> be a uniformly chosen neighbor of ρ<sup>π</sup> . Then by (6.1)

$$\mathbb{P}((G^{\pi}, X\_{\mathtt{I}}) = (H, u)) = \sum\_{v \colon \langle u, v \rangle \in E(H)} \frac{\mathbb{P}((G^{\pi}, \rho^{\pi}) = (H, v))}{\deg\_H(v)} = \frac{\sum\_{v \colon \langle u, v \rangle \in E(H)} \mathbb{P}((G, \rho) = (H, v))}{\mathbb{E}\deg(\rho)}.$$

Since <sup>ρ</sup> is uniformly distributed given <sup>G</sup>, the quantity <sup>P</sup>((G, ρ) <sup>=</sup> (H, v)) is the same for all v. So

$$\mathbb{P}((G^{\pi}, X\_{\mathsf{I}}) = (H, \mu)) = \frac{\deg\_H(\mu)\mathbb{P}((G, \rho) = (H, \mu))}{\mathbb{E}\,\mathsf{deg}(\rho)}$$

so by (6.1) the required assertion follows. 

**Corollary 6.10** *Assume that* {Gn} *is a sequence of random graphs that are almost surely finite and that* <sup>E</sup>deg(ρn) <sup>∈</sup> (0,∞) *where* ρn *is a uniformly chosen vertex of* Gn*. Let* (G<sup>π</sup> <sup>n</sup> , ρ<sup>π</sup> <sup>n</sup> ) *be the degree biasing of* (Gn, ρn)*. Assume that* Gn loc −→ (U, ρ) *and that* <sup>E</sup>deg(ρ) < <sup>∞</sup> *and that* <sup>E</sup>deg(ρn) <sup>→</sup> <sup>E</sup>deg(ρ)*. Then* <sup>G</sup><sup>π</sup> n loc −→<sup>π</sup> (U<sup>π</sup> , ρ<sup>π</sup> ) *where* (U<sup>π</sup> , ρ<sup>π</sup> ) *is the degree biasing of* (U, ρ)*. Furthermore,* (U, ρ) *and* (U<sup>π</sup> , ρ<sup>π</sup> ) *are absolutely continuous with respect to each other.*

*Conversely, assume that* {G<sup>π</sup> <sup>n</sup> } *is a sequence of random graphs that are almost surely finite and have no isolated vertices. Denote by* ρ<sup>π</sup> <sup>n</sup> *a random vertex of* G<sup>π</sup> n *drawn with probability proportional to the vertex degrees and by* (Gn, ρn) *the degree unbiasing of* (G<sup>π</sup> <sup>n</sup> , ρ<sup>π</sup> <sup>n</sup> )*. If* G<sup>π</sup> n loc −→<sup>π</sup> (U<sup>π</sup> , ρ<sup>π</sup> )*, then* Gn loc −→ (U, ρ) *where* (U, ρ) *is the degree unbiasing of* (U<sup>π</sup> , ρ<sup>π</sup> )*. Furthermore,* (U, ρ) *and* (U<sup>π</sup> , ρ<sup>π</sup> ) *are absolutely continuous with respect to each other.*

*Proof* We start by proving the first assertion. Let (H, v) be a finite rooted graph and r > 0 a fixed integer. Then by Definition 6.8

$$\mathbb{P}(\mathcal{B}\_{G\_n^{\pi}}(\rho\_n^{\pi}, r) = (H, v)) = \frac{\deg\_H(v) \mathbb{P}(\mathcal{B}\_{G\_n}(\rho\_n, r) = (H, v))}{\mathbb{E}\, \text{deg}(\rho\_n)}.$$

Since Gn loc −→ (U, ρ) and <sup>E</sup>deg(ρn) <sup>→</sup> <sup>E</sup>deg(ρ) we obtain that

$$\lim\_{n \to \infty} \mathbb{P}(B\_{G\_n^{\pi}}(\rho\_n^{\pi}, r) = (H, v)) = \frac{\deg\_H(v) \mathbb{P}(B\_U(\rho, r) = (H, v))}{\mathbb{E}\deg(\rho)} = \mathbb{P}(B\_{U^{\pi}}(\rho^{\pi}, r) = (H, v)),$$

where the last equality is also by Definition 6.8. The absolute continuity of (U, ρ) and (U<sup>π</sup> , ρ<sup>π</sup> ) follows immediately from the definition.

The second statement follows by the same proof. Note that <sup>E</sup>[deg(ρ<sup>π</sup> <sup>n</sup> )−1] → <sup>E</sup>[deg(ρ<sup>π</sup> )−1] by definition since BGπ <sup>n</sup> (ρ<sup>π</sup> <sup>n</sup> , 1) converges in distribution to BUπ (ρ<sup>π</sup> , <sup>1</sup>) and the function f ((G, ρ)) <sup>=</sup> deg(ρ)−<sup>1</sup> is a bounded continuous function on *G*•. 

We end this subsection by addressing the somewhat technical issue of verifying the condition <sup>E</sup>deg(ρn) <sup>→</sup> <sup>E</sup>deg(ρ) in Corollary 6.10. It is not guaranteed just by requiring sup<sup>n</sup> <sup>E</sup>deg(ρn) < <sup>∞</sup> as can be seen in the example of a path of length <sup>n</sup> where we choose <sup>√</sup><sup>n</sup> arbitrary vertices and add <sup>√</sup><sup>n</sup> loops to each one; in this example deg(ρ) <sup>=</sup> 2 almost surely, and <sup>E</sup>deg(ρn) <sup>=</sup> <sup>4</sup> <sup>+</sup> o(1). However, we now show that it is always possible to "truncate" the finite graphs Gn by removing edges touching vertices of large degrees so that the limit is unchanged and the average degrees converge to the expected degree of the limit. Given a finite graph G and an integer k ≥ 1 we denote by G∧k the graph obtained from G by erasing all the edges touching vertices of degree at least k. We note that even when G is connected, G∧k may be disconnected and may have isolated vertices. As we defined in Sect. 5.1, by (G ∧ k, ρ) we mean (G ∧ k[ρ],ρ) where G ∧ k[ρ] is the connected component of ρ in G ∧ k, hence it is a member of *G*• even when it is disconnected. All statements in this chapter, most importantly Corollary 6.10, do not assume the graphs involved are connected.

**Lemma 6.11** *Let* {Gn} *be a sequence of random finite graphs such that* Gn loc −→ (U, ρ) *and* <sup>E</sup>deg(ρ) < <sup>∞</sup>*. Then there exists a sequence* k(n) → ∞ *such that*

$$G\_n \land k(n) \xrightarrow{\text{loc}} (U, \rho) \dots$$

*Furthermore, if we set* G <sup>n</sup> = Gn ∧ k(n)*, then*

$$\mathbb{E}\,\mathsf{deg}\_{G\_n'}(\rho\_n) \to \mathbb{E}\,\mathsf{deg}(\rho)\,.$$

*where* ρn *is a uniformly chosen vertex of* G n*.*

*Proof* We first show that for *any* sequence k(n) → ∞ we have that Gn <sup>∧</sup> k(n) loc −→ (U, ρ). Indeed, since Gn loc −→ (U, ρ) we have that for any fixed integer r ≥ 1

$$\mathbb{P}\left(\max\left\{\deg(\upsilon) : \upsilon \in B\_{G\_n}(\rho\_n, r)\right\} \ge k(n)\right) \to 0\dots$$

If max{deg(v) : v ∈ BGn (ρn, r + 1)} < k(n), then BGn (ρn,r) = BGn∧k(n)(ρn,r). Since Gn and Gn ∧ k(n) have the same set of vertices we deduce that for any fixed r ≥ 1 and any rooted graph (H, v)

$$\mathbb{P}(B\_{G\_n \wedge k(n)}(\rho\_n, r) = (H, v)) \to \mathbb{P}(B\_U(\rho, r) = (H, v))\,.$$

Secondly, since deg(ρn) converges in distribution to deg(ρ) we have that there exists a sequence k(n) → ∞ such that <sup>E</sup>[deg(ρn) <sup>∧</sup> k(n)] → <sup>E</sup>deg(ρ). Indeed, by dominated convergence we have that <sup>E</sup>[deg(ρ)∧k] →k→∞ <sup>E</sup>deg(ρ). Furthermore, for any fixed k the function f ((G, ρ)) = deg(ρ) ∧ k is a bounded and continuous on *<sup>G</sup>*•, thus <sup>E</sup>[deg(ρn) <sup>∧</sup> <sup>k</sup>] →n→∞ <sup>E</sup>[deg(ρ) <sup>∧</sup> <sup>k</sup>]. Hence for any ε > 0 there exist <sup>k</sup> and <sup>N</sup> such that for all <sup>n</sup> <sup>≥</sup> <sup>N</sup> we have that <sup>|</sup>E[deg(ρn) <sup>∧</sup> <sup>k</sup>] − <sup>E</sup>deg(ρ)| ≤ <sup>ε</sup>. It is an exercise that this implies the existence of k(n).

Lastly, lim supEdegG n (ρn) <sup>≤</sup> <sup>E</sup>deg(ρ) since degG n (ρn) ≤ degGn (ρn) ∧ k(n). We also have that degG n (ρn) <sup>d</sup> −→ deg(ρ), hence by Fatou's lemma lim infEdegG n (ρn) <sup>≥</sup> <sup>E</sup>deg(ρ), and hence the second assertions follows. 

#### *Markings*

Given a locally convergent sequence of (possibly random) graphs Gn, we wish to apply the star-tree transform on them to create a sequence G∗ <sup>n</sup> and take its local limit of that while "remembering", in light of Lemma 6.4, the original degrees of Gn. The approach is a rather straightforward extension of the abstract setting of Sect. 5.1, see also [2]. We consider the space of triples (G, ρ, M) where G = (V , E) is a graph, <sup>ρ</sup> <sup>∈</sup> <sup>V</sup> is a vertex and <sup>M</sup> : <sup>E</sup> <sup>→</sup> <sup>R</sup> is a function assigning real values to the edges. We endow the space with a metric by setting the distance between (G1, ρ1, M1) and (G2, ρ2, M2) to be 2−<sup>R</sup> where R is the maximal value such that there exists a rooted graph isomorphism ϕ between BG<sup>1</sup> (ρ1, R) and BG<sup>2</sup> (ρ2, R) such that <sup>|</sup>M1(e) <sup>−</sup> <sup>M</sup>2(ϕ(e))| ≤ <sup>R</sup>−<sup>1</sup> for all edges <sup>e</sup> <sup>∈</sup> E(G) both of whose end points are in BG<sup>1</sup> (ρ1, R). It is easy to check that this space is again a Polish space, so again we may define convergence in distribution of random variables taking values in this space.

We say that such a random triplet (U, ρ, M) is **stationary** if conditioned on (U, ρ, M) a uniformly chosen random neighbor X<sup>1</sup> of ρ satisfies that (U, ρ, M) has the same law as (U, X1,M) in the space of isomorphism classes of rooted graphs with markings (that is, rooted isomorphisms that preserve the markings). Given a marking <sup>M</sup> we extend it to <sup>M</sup> : E(U ) <sup>∪</sup> V (U ) <sup>→</sup> <sup>R</sup> by setting M(v) <sup>=</sup> maxe:v∈<sup>e</sup> M(e) for any v ∈ V (U ). We say that (U, ρ, M) has an **exponential tail** if for some A < <sup>∞</sup> and β > 0 we have that <sup>P</sup>(M(ρ) <sup>≥</sup> s) <sup>≤</sup> Ae−βs for all <sup>s</sup> <sup>≥</sup> 0.

In the following lemma we consider a stationary triplet (U, ρ, M) that has an exponential tail and compare the hitting probabilities of certain sets when we endow the graphs with two sets of edge resistances: the first are the usual unit resistances, and in the second we may change the edge resistances arbitrarily but only on edges with high M values. We tailored the lemma this way in order to show that (G∗, Re) from Lemma 6.4 is recurrent.

**Lemma 6.12** *Let* (U, ρ, M) *be a stationary, bounded degree rooted random graph with markings which has an exponential tail. Conditioned on* (U, ρ, M) *and given some finite set* B ⊂ U*, let* **P**<sup>ρ</sup> *denote the unit-resistance random walk on* U *starting from* ρ *and let* **P** <sup>ρ</sup> *denote the random walk on* U *with edge resistances* R <sup>e</sup> *satisfying that* R <sup>e</sup> <sup>=</sup> <sup>1</sup> *whenever* M(e) <sup>≤</sup> <sup>21</sup>β−<sup>1</sup> log <sup>|</sup>B|*. Then almost surely on* (U, ρ, M) *there exists* K < ∞ *such that for any finite subset* B ⊂ U *with* |B| ≥ K *we have*

$$\left| \mathbf{P}\_{\rho} (\mathfrak{r}\_{U \backslash B} < \mathfrak{r}\_{\rho}^{+}) - \mathbf{P}\_{\rho}' (\mathfrak{r}\_{U \backslash B} < \mathfrak{r}\_{\rho}^{+}) \right| \leq \frac{1}{|B|} \cdot \varepsilon$$

*Proof* For every pair of integers T,s ≥ 1 we set

$$\mathcal{A}\_{T,s} = \left\{ \mathbf{P}\_{\rho}(\exists t \; < \; T \; : \; M(X\_t) \ge s) \le T^{\beta} e^{-\beta s/2} \right\} \dots$$

Since (U, ρ, M) is stationary and has an exponential tail for any t ≥ 0 we have

$$\mathbb{E}[\mathbf{P}\_{\rho}(M(X\_{l}) \ge s)] \le Ae^{-\beta s},$$

hence by the union bound

$$\mathbb{E}[\mathbf{P}\_{\rho}(\exists t \; ^{\prime}C \; ^{\prime}M(X\_{t}) \geq s)] \leq ATe^{-\beta s} \; .$$

Thus by Markov's inequality

$$\mathbb{P}\left(\mathcal{A}\_{T,s}^c\right) \le AT^{-2}e^{-\beta s/2}\text{ .}$$

By Borel-Cantelli we deduce that almost surely *A*T ,s occurs for all but finitely many pairs T,s. Conditioned on (U, ρ, M), we may consider only finite subsets B ⊂ U which contain ρ, since otherwise both probabilities in the statement of the lemma are 1. Let B be such a subset. By the commute time identity Lemma 2.26, and since the maximum degree of U is bounded,

$$\mathbf{E}\_{\rho}(\mathfrak{r}\_{U\backslash B}) \le C \mathcal{R}\_{\text{eff}}(\rho \leftrightarrow U \nwarrow B) |B| \le C |B|^2 \ ,$$

for some constant C > 0. The last inequality is since the resistance is bounded by |B| since there is a path of length at most |B| from ρ to U \ B. By Markov's inequality,

$$\mathbf{P}\_{\rho}(\mathbf{r}\_{U\backslash B} \ge T) \le \frac{C|B|^2}{T} \dots$$

Write S = {v ∈ U : M(v) ≥ s}. For every T,s for which *A*T ,s occurs we have

$$\mathbf{P}\_{\rho} \left( \tau\_{\mathcal{S}} < \tau\_{\{\rho\} \cup U \backslash B}^{+} \right) \le \mathbf{P}\_{\rho} (\tau\_{U \backslash B} \ge T) \; + \; \mathbf{P}\_{\rho} (\exists t < T : M(X\_t) \ge s) \le \frac{C|B|^2}{T} \; + \; T^3 e^{-\beta s/2} \; .$$

We now choose T = 2C|B| <sup>3</sup> and <sup>s</sup> <sup>=</sup> <sup>21</sup>β−<sup>1</sup> log <sup>|</sup>B<sup>|</sup> so that the right hand side of the last inequality is at most |B| <sup>−</sup><sup>1</sup> when <sup>|</sup>B<sup>|</sup> is sufficiently large. It is clear that we can couple two random walks starting from ρ, one walking on U with unit resistances and the other on (U, Re), so that they remain together until they visit a vertex of S. Hence, when |B| is large enough so that the chosen T,s are such that *A*T ,s holds we deduce from the last inequality that with probability at least 1 − |B| <sup>−</sup><sup>1</sup> the simple random walk on U visits {ρ} ∪ U \ B before visiting S, concluding our proof. 

#### **6.3 Proof of Theorem 6.1**

We now proceed to wrapping up the proof of Theorem 6.1. Recall that we have a sequence of finite planar graphs{Gn} such that Gn loc −→ (U, ρ) and with <sup>P</sup>(deg(ρ) <sup>≥</sup> k) <sup>≤</sup> Ce−ck . Our goal is to prove that (U, ρ) is almost surely recurrent.

Let us explain how we use Lemma 6.11 and Corollary 6.10 to truncate and degree bias Gn and (U, ρ) so that we may assume without loss of generality that Gn loc −→<sup>π</sup> (U, ρ). Indeed, if it does not hold that <sup>E</sup>deg(ρn) <sup>→</sup> <sup>E</sup>deg(ρ) we consider Gn∧k(n) of Lemma 6.11 which has the same limit (U, ρ). Since k(n) → ∞ the graphs Gn ∧ k(n) have non-empty set of edges (we assume that Gn have non-empty sets of edges otherwise (U, ρ) is an isolated vertex), and thus we may apply Corollary 6.10 and deduce that the degree biasing (Gn ∧ k(n), ρn) converges to the degree biasing of (U, ρ) which is absolutely continuous with respect to (U, ρ), and in particular, it is recurrent almost surely if and only if (U, ρ) is. We also erase from Gn ∧ k(n) all isolated vertices that may have been created in the truncation, since these are drawn with probability 0 after the degree bias. This will be important for us later when we unbias the graphs. Lastly, it is an easy computation using Definition 6.8 that we still have <sup>P</sup>(deg(ρ) <sup>≥</sup> k) <sup>≤</sup> Ce−ck (possibly for some other positive constants C, c). Thus, from now on we assume without loss of generality that Gn loc −→<sup>π</sup> (U, ρ) and that deg(ρ) has an exponential tail and that Gn have no isolated vertices almost surely.

Recall now the definitions and notations of Sect. 6.1. Consider the star-tree transform G∗ <sup>n</sup> of Gn and let ρ<sup>∗</sup> <sup>n</sup> be a random vertex of Tρn drawn according to the stationary distribution of Tρn . Similarly, conditioned on (U, ρ), let U<sup>∗</sup> be the star-tree transform of U and ρ<sup>∗</sup> be a random vertex of Tρ drawn according to the stationary distribution of Tρ . Furthermore, we put markings on G<sup>∗</sup> <sup>n</sup> and U<sup>∗</sup> by marking each edge e of G∗ <sup>n</sup> or U<sup>∗</sup> with deg(v) whenever e is in the tree Tv and deg(v) is the degree of v in Gn or U, respectively. Denote these markings by Mn and M, respectively.

*Claim 6.13* We have that (G∗ n, ρ<sup>∗</sup> <sup>n</sup> , Mn) for each n and (U∗, ρ∗,M) are stationary, and,

$$(G\_n^\*, \rho\_n^\*, M\_n) \stackrel{d}{\longrightarrow} (U^\*, \rho^\*, M) \dots$$

*Proof* Since for any fixed integer r > 0, the laws of BG<sup>∗</sup> <sup>n</sup> (ρ<sup>∗</sup> n,r) and BU<sup>∗</sup> (ρ∗,r) are determined by BGn (ρn,r) and BU (ρ, r), respectively, see Remark 6.3. We obtain that

$$(G\_n^\*, \rho\_n^\*, M\_n) \stackrel{d}{\longrightarrow} (U^\*, \rho^\*, M)\dots$$

Secondly, it is immediate to check that for each v ∈ Gn we have that the number of edges in Tv is precisely 2 degGn (v). This is the reason why we added the two "extra" neighbors to the root of Tv in the star tree transform described in Sect. 6.1. Thus, conditioned on Gn, for any x ∈ G<sup>∗</sup> <sup>n</sup> such that x ∈ Tv for some v ∈ Gn we have that

$$\mathbb{P}(\rho\_n^\* = \mathbf{x} \mid G\_n) = \frac{\deg\_{G\_n}(\upsilon)}{2|E(G\_n)|} \cdot \frac{\deg\_{T\_\upsilon}(\mathbf{x})}{2|E(T\_\upsilon)|} = \frac{\deg\_{T\_\upsilon}(\mathbf{x})}{2|E(G\_n^\*)|},$$

or in other words, (G∗ n, ρ<sup>∗</sup> <sup>n</sup> , Mn) is a stationary random graph and since it converges to (U∗, ρ∗,M), the latter is also stationary. 

#### **Lemma 6.14** *The triplet* (U∗, ρ∗,M) *has an exponential tail.*

*Proof* We observe that M(ρ∗) = deg(v) where v is either ρ or one of its neighbors. Hence it suffices to show that if (U, ρ) is a stationary local limit such that deg(ρ) has an exponential tail, then the random variable D(ρ) = maxv:{ρ,v}∈E(U ) deg(v) has an exponential tail. We have

$$\mathbb{P}(D(\rho) \ge k) \le \mathbb{P}(\text{deg}(\rho) \ge k) + \mathbb{P}(\text{deg}(\rho) \le k \text{ and } D(\rho) \ge k) \,. \tag{6.2}$$

The probability of the first term on the right hand side decays exponentially in k due to our assumption on (U, ρ). Conditioned on (U, ρ), let X<sup>1</sup> be a uniformly chosen random neighbor of ρ. Then clearly

$$\mathbb{P}(\deg(X\_{\mathbb{L}}) \ge k \mid \deg(\rho) \le k \text{ and } D(\rho) \ge k) \ge k^{-1}.$$

However, by stationarity <sup>P</sup>(deg(X1) <sup>≥</sup> k) <sup>=</sup> <sup>P</sup>(deg(ρ) <sup>≥</sup> k), which decays exponentially. We conclude that the second term on the right hand side of (6.2) decays exponentially as well. 

Consider the stationary random graph (U∗, ρ∗,M). By Lemma 6.14 it has an exponential tail. Consider the edge resistances

$$\mathcal{R}\_e^{\text{unit}} \equiv 1 \,, \qquad \mathcal{R}\_e^{\text{mark}} = \frac{1}{M(e)} \, .$$

In view of Lemma 6.4, it suffices to show that the network (U∗, Rmark) is almost surely recurrent, for then it will follow that U is almost surely recurrent. To prove the former, we apply the second assertion of Corollary 6.10 which allows us to assume without loss of generality that (U∗, ρ∗) is a local limit of finite planar maps (rather than a stationary local limit). In the beginning of the proof we assumed that almost surely Gn have no isolated vertices (they were erased after the degree biasing), hence the same holds for G∗ <sup>n</sup> and we may use Corollary 6.10. Since (U∗, ρ∗) is now a local limit of finite planar maps with degrees bounded by 4 we may apply Theorem 5.8 to obtain an almost sure constant c > 0 and a sequence of sets Bk ⊂ U<sup>∗</sup> such that

1. ck ≤ |Bk| ≤ <sup>c</sup>−1k, and 2. *<sup>R</sup>*eff(ρ<sup>∗</sup> <sup>↔</sup> <sup>U</sup><sup>∗</sup> \ Bk ; {Runit <sup>e</sup> }) ≥ c log k,

where we added to the conclusion of Theorem 5.8 that Bk ≥ ck since adding vertices to Bk makes the lower bound on the resistance even better.

We now define one extra set of edge resistances on U∗ which will allow us to interpolate between the edge resistances <sup>R</sup>unit and <sup>R</sup>mark. For each integer <sup>k</sup> <sup>≥</sup> 1 we define

$$R\_e^{\text{mid}} = \begin{cases} 1 & M(e) \le C \log k, \\ M^{-1}(e) & \text{otherwise}, \end{cases}$$

where C > 0 is some large constant that will be chosen later. We will use **P**, **P**mark and **P**mid to denote the probability measures, conditioned on (U∗, ρ∗,M), of random walks on <sup>U</sup><sup>∗</sup> with edge resistances{Runit <sup>e</sup> }, {Rmark <sup>e</sup> } and {Rmid <sup>e</sup> }, respectively.

**Lemma 6.15** *For some other constant* c > 0 *we have*

$$\mathcal{R}\_{\text{eff}}(\rho^\* \leftrightarrow U^\* \backslash B\_k \; ; \; \{R\_e^{\text{mod}}\}) \ge c \log k \; .$$

*Proof* We may assume k is large enough so that M(e) ≤ C log k for every edge e incident to ρ∗. By Claim 2.22 we have

$$\mathcal{R}\_{\text{eff}}(\rho^\* \leftrightarrow U^\* \backslash B\_k \; ; \; \{ \mathcal{R}\_\varepsilon^{\text{unit}} \}) \le \frac{1}{\mathbf{P}\_{\rho^\*} (\tau\_{U^\* \backslash B\_k} < \tau\_{\rho^\*}^+)} \; ,$$

hence

$$\mathbf{P}\_{\rho^\*} (\mathfrak{r}\_{U^\* \backslash B\_k} < \mathfrak{r}\_{\rho^\*}^+) \le \frac{1}{c \log k},$$

by our assumption on Bk above. By Lemma 6.12 it follows that

$$\mathbf{P}\_{\rho^\*}^{\mathrm{mid}}(\mathfrak{r}\_{U^\*\backslash B\_k} < \mathfrak{r}\_{\rho^\*}^+) \le \frac{2}{c \log k},$$

when <sup>k</sup> is large enough and the constant C > 0 in the definition of {Rmid <sup>e</sup> } is chosen large enough with respect to β. Using Claim 2.22 again and the fact that U∗ has degrees bounded by 4 concludes the proof. 

We need yet another easy general fact about electric networks.

*Claim 6.16* Consider a finite network G in which all resistances are bounded above by 1. Then for any integer m ≥ 1 and any two vertices a = z we have

$$\mathcal{R}\_{\rm eff}(B\_G(a,m)\leftrightarrow z) \ge \mathcal{R}\_{\rm eff}(a\leftrightarrow z) - m\ .$$

*Proof* Let <sup>θ</sup><sup>m</sup> be the unit current flow from B(a, m) to <sup>z</sup>. For a vertex <sup>v</sup> <sup>∈</sup> B(a, m) denote

$$\alpha\_v = \sum\_{u \notin B(a,m): u \sim v} \theta^m(vu).$$

so that αv ≥ 0 for all v ∈ B(a, m) and <sup>v</sup>∈B(a,m) αv <sup>=</sup> 1. For a vertex <sup>v</sup> <sup>∈</sup> B(a, m) let θ a,v be a unit flow putting flow 1 on some shortest path from a to v in B(a, m). Set

$$\theta = \sum\_{v \in B(a,m)} \alpha\_v (\theta^m + \theta^{a,v}) \dots$$

 By Thomson's principle (Theorem 2.28), Jensen's inequality and since <sup>v</sup> αv = 1 we have

$$\mathcal{R}\_{\text{eff}}(a \leftrightarrow z) \le \mathcal{E}(\theta) = \mathcal{E}(\theta^{\text{uf}}) + \sum\_{\epsilon} r\_{\epsilon} \left[ \sum\_{v \in B(a,m)} a\_{v} \theta^{a,v}(\epsilon) \right]^{2} \le \mathcal{E}(\theta^{\text{uf}}) + \sum\_{v \in B(a,m)} a\_{v} \sum\_{\epsilon} r\_{\epsilon} \left( \theta^{a,v}(\epsilon) \right)^{2}$$

$$\le \mathcal{E}(\theta^{\text{uf}}) + \sum\_{v \in B(a,m)} a\_{v} \cdot m = \mathcal{R}\_{\text{eff}}(B(a,m) \leftrightarrow z) + m \text{ .} \tag{7}$$

We are finally ready to conclude the proof of the main theorem of this chapter.

*Proof of Theorem 6.1* By Lemma 6.15 and Claim 6.16 we have that the sets Bk obtained earlier satisfy that for any m ≥ 0

$$\mathcal{R}\_{\mathrm{eff}}(B\_{U^\*}(\rho^\*,m) \leftrightarrow U^\* \backslash B\_k \; ; \; \{R\_e^{\mathrm{mid}}\}) \ge c \log k - m \dots$$

Moreover, for every edge e,

$$
\mathcal{R}\_e^{\mathrm{mark}} \ge \frac{\mathcal{R}\_e^{\mathrm{mid}}}{C \log k},
$$

hence

$$\mathcal{R}\_{\mathrm{eff}}(B\_{U^\*}(\rho^\*,m) \leftrightarrow U^\* \backslash B\_k \; ; \; \{R\_e^{\mathrm{mark}}\}) \ge c/C - m/C \log k\dots$$

By taking k → ∞ we deduce that there exists c > 0 such that for any m ≥ 1

$$\mathcal{R}\_{\text{eff}}(B\_{U^\*}(\rho^\*,m) \leftrightarrow \infty; \{R\_e^{\text{mark}}\}) \ge c \dots$$

Consider the current unit flow from <sup>ρ</sup><sup>∗</sup> to <sup>∞</sup> in (U∗,{Rmark <sup>e</sup> }). If this flow had finite energy, then for any ε > 0 there would exists m ≥ 1 such that *R*eff(BU<sup>∗</sup> (ρ∗, m) ↔ ∞; {Rmark <sup>e</sup> }) ≤ ε, which is a contradiction to the above. Hence

$$\mathcal{R}\_{\text{eff}}(\rho^\* \leftrightarrow \infty; \{R\_e^{\text{mark}}\}) = \infty \,,$$

that is, (U∗,{Rmark <sup>e</sup> }) is almost surely recurrent. The theorem now follows by Lemma 6.4.  **Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 7 Uniform Spanning Trees of Planar Graphs**

#### **7.1 Introduction**

Let G be a finite connected graph. A **spanning tree** T of G is a connected subgraph of G that contains no cycles and such that every vertex of G is incident to at least one edge of T . The set of spanning trees of a given finite connected graph is obviously finite and hence we may draw one uniformly at random. This random tree is called the **uniform spanning tree** (UST) of G. This model was first studied by Kirchhoff [49] who gave a formula for the number of spanning trees of a given graph and provided a beautiful connection with the theory of electric networks. In particular, he showed that the probability that a given edge {x,y} of G is contained in the UST equals *R*eff(x ↔ y; G); we prove this fundamental formula in Sect. 7.2 (see Theorem 7.2).

Is there a natural way of defining a UST probability measure on an infinite connected graph? It will soon become clear that we have set the framework already in Sect. 2.3 to answer this question positively. Let G = (V , E) be an infinite connected graph and assume that {Gn} is a finite exhaustion of G as defined in Sect. 2.5. That is, {Gn} is a sequence of finite graphs, Gn ⊂ Gn+<sup>1</sup> for all n, and ∪Gn = G. Russell Lyons conjectured that the UST probability measure on Gn converges weakly to some probability measure on subsets of E and in his pioneering work Pemantle [68] showed that it is indeed the case.

More precisely, denote by *T*<sup>n</sup> a UST of Gn, then it is shown in [68] that for any two finite subset of edges A,B of G the limit

$$\lim\_{n \to \infty} \mathbf{P}(A \subset \mathcal{T}\_n \; , \; B \cap \mathcal{T}\_n = \emptyset) \; , \tag{7.1}$$

exists and does not depend on the exhaustion {Gn}. The proof is a consequence of Rayleigh's monotonicity (Corollary 2.29) and will be presented in Sect. 7.3. This together with Kolmogorov's extension theorem [24, Theorem A.3.1] implies that there exists a unique probability measure on infinite subsets of E for which a sample

<sup>©</sup> The Author(s) 2020

A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes

of F satisfies

$$\mathbf{P}(A \subset \mathfrak{F} \; , \; B \cap \mathfrak{F} = \emptyset) = \lim\_{n \to \infty} \mathbf{P}(A \subset \mathcal{T}\_n \; , \; B \cap \mathcal{T}\_n = \emptyset) \; ,$$

for any two finite subsets of edges A and B of G. Thus, the law of F is determined and we denote it by μ<sup>F</sup> . The superscript F stands for *free* and will be explained momentarily. Let us explore some properties of μ<sup>F</sup> that are immediate from its definition.

Since every vertex of G is touched by at least one edge of *T*<sup>n</sup> with probability 1 when n is large enough (so that Gn contains the vertex), we learn that the edges of F almost surely touch every vertex of G, that is, F is almost surely *spanning*. Similarly, the probability that the edges of a given cycle in G are contained in *T*<sup>n</sup> (once n is large enough so that Gn contains the cycle) is 0. Since G has countably many cycles we deduce that almost surely there are no cycles in F. By a similar reasoning we deduce that almost surely any connected component of F is infinite. However, a moment's reflection shows that this kind of reasoning cannot be used to determine that F is almost surely connected.

It turns out, perhaps surprisingly, that F need not be connected almost surely. A remarkable result of Pemantle [68] shows that a sample of μ<sup>F</sup> on Z<sup>d</sup> is almost surely connected when d = 1, 2, 3, 4 and almost surely disconnected when d ≥ 5. Since it may be the case that a sample of μ<sup>F</sup> is disconnected with positive probability, we call μ<sup>F</sup> the **free uniform spanning forest** (rather than tree) of G, denoted henceforth FUSFG. The term *free* corresponds to the fact that we have not imposed any boundary conditions when taking a limit. It will be very useful to take other boundary conditions, such as the *wired* boundary condition, see Sect. 7.3. The seminal paper of Benjamini et al. [12] explores many properties of these infinite random trees (properties such as number of components and connectivity in particular, size of the trees, recurrence or transience of the trees and many others) on various underlying graphs with an emphasis on Cayley graphs. We refer the reader to [12] and to [61, Chapters 4 and 10] for a comprehensive treatment.

The question of connectivity of the FUSF is therefore fundamental and unfortunately it is not even known that connectivity is an event of probability 0 or 1 on any graph G, see [12, Question 15.7]. In [44] the circle packing theorem (Theorem 3.5) is used to prove that FUSF<sup>G</sup> is almost surely connected when G is a bounded degree proper planar map, answering a question of [12, Question 15.2]. Our goal in this chapter is to present a proof for a specific case where G is a bounded degree, transient, one-ended planar triangulations. Even though this is a particular case of a general theorem, the argument we present here contains most of the key ideas. We refer the interested reader to [44] for the general statement.

**Theorem 7.1 ([44])** *Let* G *be a simple, bounded degree, transient, one-ended planar triangulation. Then* FUSF<sup>G</sup> *is almost surely connected.*

The rest of this chapter is organized as follows. In Sect. 7.2 we discuss two basic properties of USTs on finite graphs. Namely, Kirchhoff's effective resistance formula mentioned earlier and the spatial Markov property for the UST. In Sect. 7.3 we prove Pemantle's [68] result (7.1) showing that FUSF<sup>G</sup> exists. We will also define there the *wired uniform spanning forest* which is obtained by taking a limit of the UST probability measures over exhaustions with wired boundary. We will also need some fairly basic notions of electric networks on infinite graphs that we have not discussed in Sect. 2.5. Next, in Sect. 7.4 we will restrict to the setting of planar graph and employ planar duality to obtain an extremely useful connection between the free and wired spanning forests which will be useful later. Using these tools we have collected we will prove Theorem 7.1 in Sect. 7.5.

#### **7.2 Basic Properties of the UST**

#### *Kirchhoff's Effective Resistance Formula*

**Theorem 7.2 (Kirchoff [49])** *Let* G *be a finite connected graph and denote by T a uniformly drawn spanning tree of* G*. Then for any edge* e = (x, y) *we have*

$$\mathbf{P}(e \in \mathcal{T}) = \mathcal{R}\_{\text{eff}}(\mathbf{x} \leftrightarrow \mathbf{y})\,.$$

*Proof* Let a = z be two distinct vertices of G (later we will take a = x and z = y) and note that any spanning tree of G contains precisely one path connecting a and z. Thus, a uniformly drawn spanning tree induces a random path from a to z. By Claim 2.46 we obtain a unit flow θ from a to z. To be concrete, for each edge e we have that θ (e) is the probability that the random path from a to z traverses e minus the probability that it traverses e. We will now show that θ satisfies the cycle law (see Claim 2.14), so it is in fact the unit current flow (see Definition 2.19).

Let e1,..., e <sup>m</sup> be a directed cycle in G. Our goal is to show that

$$\sum\_{l=1}^{m} \theta(\vec{e\_l}) = 0 \,. \tag{7.2}$$

Denote by T (G) the set of spanning trees of G. Expanding the sum on the left hand side with the definition of θ we get that it equals

$$\left| |T(G)|^{-1} \sum\_{t \in T(G)} \sum\_{i=1}^{m} f\_i^+(t) - |T(G)|^{-1} \sum\_{t \in T(G)} \sum\_{j=1}^{m} f\_j^-(t) \right|,$$

where f + <sup>i</sup> (t) equals 1 if the unique path from a to z in t traverses e<sup>i</sup> and 0 otherwise, and similarly, f − <sup>j</sup> (t) equals 1 if this path traverses ejand 0 otherwise.

For 1 ≤ i ≤ m we denote by T <sup>+</sup> <sup>i</sup> the set of pairs (t, i) for which f <sup>+</sup> <sup>i</sup> (t) = 1. Similarly define T − <sup>j</sup> as the set of pairs (t, j ) for which f <sup>−</sup> <sup>j</sup> (t) = 1. To prove (7.2) it suffices to show that

$$|\uplus\_{l \in \{1, ..., m\}} T\_j^+| = |\uplus\_{j \in \{1, ..., m\}} T\_j^-| \, .$$

Let (t, i) ∈ T <sup>+</sup> <sup>i</sup> . The graph t \ {ei} has two connected components. Let e<sup>j</sup> be the first edge after ei, in the order of the cycle e1,..., e m, that is incident to both connected components and consider the spanning tree t = t ∪ {ej }\{ei}. Note that the unique path in t from a to z traverses ej, so (t ,j) ∈ T <sup>−</sup> <sup>j</sup> . This procedure defines a bijection from "iT <sup>+</sup> <sup>i</sup> to "<sup>j</sup> T <sup>−</sup> <sup>j</sup> . Indeed, given (t ,j) from before, we can erase ej and go on the cycle in the opposite order until we reach ei which has to be the first edge incident to the two connected components of t \ {ej }. This shows (7.2) and concludes the proof. 

#### *Spatial Markov Property of the UST*

We would like to study the UST probability measure conditioned on the event that some edges are present in the UST and others not. It turns out that sampling from this conditional distribution amounts to drawing a UST on a modified graph.

Let G = (V , E) be a finite connected graph and let A and B be two disjoint subsets of edges. We write (G − B)/A for the graph obtained from G by erasing the edges of B and contracting the edges of A. We identify the edges of (G − B)/A with the edges E \ B. Denote by *T*<sup>G</sup> and *T*(G−B)/A a UST on G and (G − B)/A, respectively, and assume that

$$\mathbf{P}(A \subset \mathcal{T}\_G \,, \ B \cap \mathcal{T}\_G = \emptyset) > 0 \dots$$

This assumption is equivalent to G − B being connected and that A contains no cycles.

Then, conditioned on the event that *T*<sup>G</sup> contains the edges A and does not contain any edge of B the distribution of *T*<sup>G</sup> is equal to the union of A with *T*(G−B)/A. In other words, for a set *A* of spanning trees of G we have that

$$\mathbf{P}(\mathcal{T}\_{\mathbb{G}} \in \mathcal{A} \mid \mathcal{A} \subset \mathcal{T}\_{\mathbb{G}}, \; \mathcal{B} \cap \mathcal{T}\_{\mathbb{G}} = \emptyset) = \mathbf{P}(\mathcal{A} \cup \mathcal{T}\_{(G-\mathcal{B})/A} \in \mathcal{A}') \;. \tag{7.3}$$

The proof of (7.3) follows immediately from the observation that the set of spanning trees of G not containing any edge of B is simply the set of spanning trees of G−B. Similarly, the set of spanning trees of G containing all the edges of A is simply the union of A to each spanning tree of G/A, and (7.3) follows.

#### **7.3 Limits over Exhaustions: The Free and Wired USF**

Let G be an infinite connected graph and let {Gn} be a finite exhaustion of it. In this section we will show that (7.1) holds and that the UST measures with *wired* boundary conditions also converge. Let us first explain the latter. Denote by G∗ <sup>n</sup> the graph obtained from G by identifying the infinite set of vertices G \ Gn to a single vertex zn and erasing the loops at zn formed by this identification. We say that {G<sup>∗</sup> n} is a **wired finite exhaustion** of G.

**Theorem 7.3 (Pemantle [68])** *Let* G *be an infinite connected graph,* {Gn} *a finite exhaustion and* {G<sup>∗</sup> <sup>n</sup>} *the corresponding wired finite exhaustion. Denote by T*<sup>n</sup> *and T* ∗ <sup>n</sup> *USTs on* Gn *and* G<sup>∗</sup> <sup>n</sup>*, respectively. Then for any two finite disjoint subsets* A,B ⊂ E(G) *of edges of* G *we have that the limits*

$$\lim\_{n \to \infty} \mathbf{P}(A \subset \mathcal{T}\_n \, , \, \, B \cap \mathcal{T}\_n = \emptyset) \, ,$$

*and*

$$\lim\_{n \to \infty} \mathbf{P}(A \subset \mathcal{T}\_n^\* \,, \ B \cap \mathcal{T}\_n^\* = \emptyset) \,,$$

*exist and do not depend on the exhaustion* {Gn}*.*

We postpone the proof for a little longer and first discuss some of its implications. As mentioned earlier, Theorem 7.3 together with Kolmogorov's extension theorem [24, Theorem A.3.1] implies that there exists two probability measures μ<sup>F</sup> and μ<sup>W</sup> on infinite subsets of the edges of E arising as the unique limits of the laws *T*<sup>n</sup> and *T* ∗ <sup>n</sup> . That is, the samples F<sup>f</sup> and F<sup>w</sup> of μ<sup>F</sup> and μ<sup>W</sup> satisfy

$$\mathbf{P}(A \subset \mathfrak{F}^f, \ B \cap \mathfrak{F}^f = \emptyset) = \lim\_{n \to \infty} \mathbf{P}(A \subset \mathcal{T}\_n, \ B \cap \mathcal{T}\_n = \emptyset)\,,$$

and

$$\mathbf{P}(A \subset \mathfrak{F}^w, \ B \cap \mathfrak{F}^w = \emptyset) = \lim\_{n \to \infty} \mathbf{P}(A \subset \mathcal{T}\_n^\*, \ B \cap \mathcal{T}\_n^\* = \emptyset) \dots$$

We call μ<sup>F</sup> and μ<sup>W</sup> the **free uniform spanning forest** and the **wired uniform spanning forest** and denote them by FUSF<sup>G</sup> and WUSF<sup>G</sup> respectively. We have seen earlier (one paragraph below (7.1)) that both F<sup>f</sup> and F<sup>w</sup> are almost surely spanning forests, that is, spanning graphs of G with no cycles and that every connected component of them is infinite. Thus μ<sup>F</sup> and μ<sup>W</sup> are supported on what are known as **essential spanning forests** of G, that is, spanning forests of G in which every component is infinite.

Are the probability measures FUSF<sup>G</sup> and WUSF<sup>G</sup> equal? Not necessarily. It is easy to see that on the infinite path Z the WUSF<sup>Z</sup> and the FUSF<sup>Z</sup> are equal and are the entire graph Z with probability 1. Conversely, it is not very difficult to see that they are different on a 3-regular tree, see exercise 1 of this chapter. Pemantle [68] has shown that FUSFZ<sup>d</sup> = WUSFZ<sup>d</sup> for any d ≥ 1 and a very useful criterion for determining whether there is equality was developed in [12]. We refer the reader to [61, Chapter 10] for further reading.

Before presenting the proof of Theorem 7.3 let us make a few short observations regarding the effective resistance between two vertices in an infinite graph, extending what we proved in Sect. 2.5.

#### **Effective Resistance in Infinite Networks**

Let G be an infinite connected graph. We have seen in Sect. 2.5 that for any vertex v the electric resistance *R*eff(v ↔ ∞) from v to ∞ is well defined as the limit of *R*eff(a ↔ zn; G<sup>∗</sup> n) where {G<sup>∗</sup> <sup>n</sup>} is a wired finite exhaustion and zn is the vertex resulting in the identification of the vertices G \ Gn.

To define the electric resistance between two vertices v, u of an infinite graph, one has to take exhaustions and specify boundary conditions since the limits may differ depending on them.

*Claim 7.4* Let G be an infinite connected graph, {Gn} a finite exhaustion and {G<sup>∗</sup> n} a wired finite exhaustion. Then for any two vertices u, v of G we have that the limits

$$\mathcal{R}^{F}\_{\text{eff}}(\mu \leftrightarrow v; G) := \lim\_{n} \mathcal{R}\_{\text{eff}}(\mu \leftrightarrow v; G\_n) \,,$$

and

$$\mathcal{R}\_{\mathrm{eff}}^{W}(\mu \leftrightarrow v; G) := \lim\_{n} \mathcal{R}\_{\mathrm{eff}}(\mu \leftrightarrow v; G\_n^\*)\,,$$

exist and do not depend on the exhaustion {Gn}.

*Proof* For the first limit we note that by Rayleigh's monotonicity (Corollary 2.29), the sequence *R*eff(u ↔ v; Gn) is non-increasing and non-negative since Gn ⊂ Gn+1, hence it converges. A sandwiching argument as in the proof of Claim 7.4 shows that the limit does not depend on the exhaustion {Gn}.

For the second limit, since Gn can be obtained by gluing vertices of Gn+<sup>1</sup> we deduce by Corollary 2.30 that the sequence *R*eff(u ↔ v; G<sup>∗</sup> n) is non-decreasing and bounded (by the graph distance in G between u and v for instance), hence it converges. The limit does not depend on the exhaustion by an identical sandwiching argument. 

We call *<sup>R</sup>*<sup>F</sup> eff(u <sup>↔</sup> <sup>v</sup>; G) and *<sup>R</sup>*<sup>W</sup> eff(u ↔ v; G) the **free effective resistance** and **wired effective resistance** between u and v respectively.

#### *Proof of Theorem 7.3*

We will prove the assertion regarding the first limit; the second is almost identical. Write A = {e1,...,ek} and ei = (xi, yi) for each 1 ≤ i ≤ k. Assume without loss of generality that Gn contains A for all n. As before, denote by *T*<sup>n</sup> a UST of Gn. By (7.3) and Theorem 7.2 we have that

$$\mathbf{P}(A \subset \mathcal{T}\_n) = \prod\_{i=1}^k \mathbf{P}(e\_i \in \mathcal{T}\_n \mid e\_j \in \mathcal{T}\_n \quad \forall \ j < i) = \prod\_{i=1}^k \mathcal{R}\_{\text{eff}}(\mathbf{x}\_i \leftrightarrow \mathbf{y}\_i; G\_n / \{e\_1, \dots, e\_{i-1}\}) \dots$$

Note that  Gn/{e1,...,ei−1} is a finite exhaustion of the infinite graph G/{e1,...,ei−1} and so by Claim 7.4 we obtain that the limit

$$\lim\_{n} \mathbf{P}(A \subset \mathcal{T}\_n) = \prod\_{i=1}^k \mathcal{R}\_{\text{eff}}(\mathbf{x}\_i \leftrightarrow \mathbf{y}\_i; G/\{e\_1, \dots, e\_{l-1}\}),$$

exists and does not depend on the exhaustion.

Since we know this limit exists for all finite edge sets A, it follows by the inclusion-exclusion formula that **P**(A ⊂ *T*n, B ∩ *T*<sup>n</sup> = ∅) converges for any finite sets A,B, concluding our proof. 

It is now quite pleasant to see that the symbiotic relationship between electric network and UST theories continues to flourish in the infinite setting. Indeed, by combining Theorems 7.3 and Claim 7.4 we obtain the extension of Kirchhoff's formula for infinite connected graphs.

**Theorem 7.5** *Let* G *be an infinite connected graph and denote by* F<sup>F</sup> *and* F<sup>W</sup> *a sample from* FUSF<sup>G</sup> *and* WUSF<sup>G</sup> *respectively. Then for any edge* e = (x, y) *of* G *we have that*

$$\mathbf{P}(e \in \mathfrak{F}^F) = \mathcal{R}\_{\mathrm{eff}}^F(\mathfrak{x} \leftrightarrow \mathfrak{y}; G),$$

*and*

$$\mathbf{P}(e \in \mathfrak{F}^W) = \mathcal{R}\_{\text{eff}}^W(\mathfrak{x} \leftrightarrow \mathfrak{y}; G)\,.$$

#### **7.4 Planar Duality**

When G is planar there is a very useful relationship between FUSF<sup>G</sup> and WUSFG. Recall that given a planar map G, the **dual graph** of G is the graph G† whose vertex set is the set of faces of G and two faces are adjacent in G† if they share an edge in G. Thus, G† is locally-finite if and only if every face of G has finitely many edges. To each edge <sup>e</sup> <sup>∈</sup> E(G) corresponds a dual edge <sup>e</sup>† <sup>∈</sup> E(G†) which is the pair of faces of G incident to e; this is clearly a one-to-one correspondence.

When G is a finite planar graph, this correspondence induces a one-to-one correspondence between the set of spanning trees of G and the set of spanning trees of G†. Given a spanning tree of t of G we slightly abuse the notation and write t † for the set of edges {e† : <sup>e</sup> <sup>∈</sup> <sup>G</sup> \ <sup>t</sup>}, that is

$$e \in t \iff e^{\dagger} \notin t^{\dagger} \dots$$

If t † has a cycle, then t is disconnected. Furthermore, if there is a vertex G† not incident to any edge of t †, then all the edges of the corresponding face in G are present in t hence t contains a cycle. We deduce that if t is a spanning tree of G, then t † is a spanning tree of <sup>G</sup>†. The converse also holds since (t†)† <sup>=</sup> <sup>t</sup> and (G†)† <sup>=</sup> <sup>G</sup>.

Now assume that G is an infinite planar maps such that G† is locally finite. Given an essential spanning forest F of G we similarly define F† as the set of edges {e† : <sup>e</sup> <sup>∈</sup> <sup>G</sup> \ <sup>F</sup>}. A similar argument shows that <sup>F</sup>† is an essential spanning forest of G†. This raises the natural question: when F is a sample of FUSFG, what is the law of F†? The answer in general is an object known as the *transboundary uniform spanning forest* [44, Proposition 5.1]. However, when G is additionally assumed to be one-ended (in particular, in the setting of Theorem 7.1) it turns out that F† is distributed as WUSFG† :

**Proposition 7.6** *Let* G *be an infinite, one-ended planar map with a locally finite dual* G† *and let* F *be a sample of* FUSFG*. Then the law of* F† *is* WUSFG† *.*

*Proof* Let Gn be a finite exhaustion of G. Let Fn be a finite exhaustion G† defined by letting <sup>f</sup> <sup>∈</sup> Fn if and only if every vertex of <sup>f</sup> in <sup>G</sup> belongs to Gn. Then <sup>G</sup>† n is obtained from <sup>G</sup>† by contracting <sup>G</sup>† \ Fn into a single vertex which corresponds to the outer face of Gn. Thus, G† <sup>n</sup> is a wired exhaustion of G† and the statement follows. 

We use to obtain an important criterion of connectivity of FUSF<sup>G</sup> in the planar case.

**Proposition 7.7** *Let* G *be an infinite, one-ended planar map with a locally finite dual* G†*. Then a sample of* FUSF<sup>G</sup> *is connected almost surely if and only if each component of a sample of* WUSF<sup>G</sup> *is one-ended almost surely.*

*Proof* By Proposition 7.6 it suffices to show that if F is an essential spanning forest of G, then F is connected if and only if every component of F† is one-ended. Indeed, if F is disconnected, then the boundary of a connected component of F induces an bi-infinite path in F†. Conversely, if F† contains a bi-infinite path, then by the Jordan curve theorem F is disconnected. 

#### **7.5 Connectivity of the Free Forest**

#### *Last Note on Infinite Networks*

We make two more useful and natural definitions. Given two disjoint finite sets A and B in an infinite connected graph G we define the free and wired effective resistance between them *<sup>R</sup>*<sup>W</sup> eff(A <sup>↔</sup> <sup>B</sup>; G) and *<sup>R</sup>*<sup>F</sup> eff(A ↔ B; G) as the free and wired effective resistance between a and b in the graph obtained from G by identifying A and B to the vertices a and b.

Lastly, given a graph G, a wired finite exhaustion {G<sup>∗</sup> <sup>n</sup>} of G and two disjoint finite sets A and B we define

$$\mathcal{R}\_{\text{eff}}(A \leftrightarrow B \cup \{\infty\}; G) := \lim\_{n \to \infty} \mathcal{R}\_{\text{eff}}(A \leftrightarrow B \cup \{z\_n\}; G\_n^\*), \tag{7.4}$$

where the last limit exists since the sequence is non-increasing from n that is large enough so that Gn contains A and B. In the proof of Theorem 7.1 we will require the following estimate.

**Lemma 7.8** *Let* A *and* B *be two finite sets of vertices in an infinite connected graph* G*. Then*

*R*W eff(A ↔ B; G) ≤ 3 max *R*eff (A ↔ B ∪ {∞}; G), *R*eff (B ↔ A ∪ {∞}; G) .

*Proof* For any three distinct vertices u, v, w in a finite network we have by the union bound that **P**u(τ{v,w} < τ <sup>+</sup> <sup>u</sup> ) ≤ **P**u(τv < τ <sup>+</sup> <sup>u</sup> ) + **P**u(τw < τ <sup>+</sup> <sup>u</sup> ). Hence by Claim 2.22 we get that

$$\mathcal{R}\_{\text{eff}}(\boldsymbol{\mu}\leftrightarrow\{\boldsymbol{v},\boldsymbol{w}\})^{-1}\leq\mathcal{R}\_{\text{eff}}(\boldsymbol{\mu}\leftrightarrow\boldsymbol{v})^{-1}+\mathcal{R}\_{\text{eff}}(\boldsymbol{\mu}\leftrightarrow\boldsymbol{w})^{-1}\ .$$

Let {G<sup>∗</sup> <sup>n</sup>} be a wired finite exhaustion of G and assume without loss of generality that A and B are contained in G∗ <sup>n</sup> for all n. Then by the previous estimate

$$\mathcal{R}\_{\text{eff}}(A \leftrightarrow B \cup \{z\_n\}; G\_n^\*)^{-1} \le \mathcal{R}\_{\text{eff}}(A \leftrightarrow B; G\_n^\*)^{-1} + \mathcal{R}\_{\text{eff}}(A \leftrightarrow z\_n; G\_n^\*)^{-1}.$$

Denote by M the maximum in the statement of the lemma and take n → ∞ in the last inequality. We obtain that

$$M^{-1} \le \mathcal{R}\_{\text{eff}}(A \leftrightarrow B \cup \{\infty\}; G)^{-1} \le \mathcal{R}\_{\text{eff}}^W(A \leftrightarrow B; G)^{-1} + \mathcal{R}\_{\text{eff}}(A \leftrightarrow \infty; G)^{-1} \dots$$

Rearranging gives that

$$
\mathcal{R}\_{\rm eff}(A \leftrightarrow \infty; G) \le \frac{M \mathcal{R}\_{\rm eff}^W(A \leftrightarrow B; G)}{\mathcal{R}\_{\rm eff}^W(A \leftrightarrow B; G) - M}.
$$

By symmetry, the same inequality holds when we replace the roles of A and B. We put this together with the triangle inequality for effective resistances (2.9) and get that

$$\mathcal{R}\_{\text{eff}}^{W}(A \leftrightarrow B; \ G) \leq \mathcal{R}\_{\text{eff}}(A \leftrightarrow \infty; G) \ + \ \mathcal{R}\_{\text{eff}}(B \leftrightarrow \infty; G) \leq \frac{2M\mathcal{R}\_{\text{eff}}^{W}(A \leftrightarrow B; G)}{\mathcal{R}\_{\text{eff}}^{W}(A \leftrightarrow B; G) - M}.$$

which by rearranging gives the desired inequality. 

#### *Method of Random Sets*

We present the following weakening of the method of random paths as in Sect. 2.6. Let μ be the law of a random subset W of vertices of G. Define the *energy* of μ as

$$\mathcal{E}(\mu) = \sum\_{v \in V} \mu(v \in W)^2.$$

**Lemma 7.9 (Method of Random Sets)** *Let* A,B *be two disjoint finite sets of vertices in an infinite graph* G*. Let* W *be a random subset of vertices of* G *and denote by* μ *its law. Assume that the subgraph of* G *induced by* W *almost surely contains a simple path starting at* A *that is either infinite or finite and ends at* B*. Then*

$$\mathcal{R}\_{\text{eff}}(A \leftrightarrow B \cup \{\infty\}; G) \le \mathcal{E}(\mu). \tag{7.5}$$

*Proof* Given W let γ be a simple path, contained in W, connecting A to B or an infinite path starting at A. We choose γ according to some prescribed lexicographical ordering. Then, letting ν be the law of γ ,

$$\mathcal{E}(\nu) \le \sum\_{\vec{e} \in E} \nu(\vec{e} \in \nu)^2,$$

where by e ∈ γ we mean that the directed edge e is traversed (in its direction) by γ , and by *E*(ν) we mean the energy of the flow induced by γ , as in Claim 2.46.

Let γ be an independent random path having the same law as γ . Then the sum above is precisely the expected number of directed edges traversed both by γ and γ . Since these are simple paths, they each contain at most one directed edge emanating from each vertex v ∈ W. Thus, the expected number of directed edges used by both paths is at most the number of vertices used by both paths. Hence,

$$\mathcal{E}(\boldsymbol{\nu}) \le \sum\_{\boldsymbol{v} \in V(G)} \boldsymbol{\nu}(\boldsymbol{v} \in \boldsymbol{\nu})^2 \le \sum\_{\boldsymbol{v} \in V(G)} \mu(\boldsymbol{v} \in W)^2 = \mathcal{E}(\boldsymbol{\mu})\,,$$

and the proof is concluded by Thomson's principle (Theorem 2.28). 

#### *Proof of Theorem 7.1*

In Theorem 7.1 we assume that G = (V , E) is a bounded-degree, one-ended triangulation. Hence G† is a bounded degree (in fact, 3-regular), one-ended and transient planar map with faces of uniformly bounded size. We leave this verification as an exercise for the reader. To avoid carrying the † symbol around, and with a slight abuse of notation, let G = (V , E) be a graph satisfying these assumptions on G†, that is, we assume that G is a one-ended, transient, infinite planar map with bounded degrees and face sizes. We will prove under these assumptions that every component of WUSF<sup>G</sup> is one ended almost surely which implies Theorem 7.1 by Proposition 7.7.

Let T be the bounded-degree one-ended triangulation obtained from G by adding a vertex inside each face of G and connecting it by edges to the vertices of that face according to their cyclic ordering. By Theorem 4.4 there exists a circle packing of T in the unit disc U. We identify the vertices of T as the vertices V (G) and faces F (G) of G, and denote this circle packing as P = {P (v) : v ∈ V (G)}∪{P (f ) : f ∈ F (G)}.

Given <sup>z</sup> <sup>∈</sup> <sup>U</sup> and <sup>r</sup> <sup>≥</sup> r > 0 denote by Az(r, r ) the annulus {<sup>w</sup> <sup>∈</sup> <sup>C</sup> : <sup>r</sup> <sup>≤</sup> |w − z| ≤ r }.

**Definition 7.10** Write Vz(r, r ) for the set of vertices v of G such that either


We emphasize that Vz(r, r ) contains only vertices of G; no vertices of T that correspond to faces of G belong to it.

**Lemma 7.11** *There exists a constant* C < ∞ *depending only on the maximal degree such that for any* <sup>z</sup> <sup>∈</sup> <sup>U</sup> *and any positive integer* <sup>n</sup> *satisfying* <sup>|</sup>z| ≥ <sup>1</sup> <sup>−</sup> <sup>C</sup>−<sup>n</sup> *the sets*

$$V\_{\mathfrak{z}}(\mathbb{C}^{-i}, \mathfrak{z}\mathbb{C}^{-i}) \qquad 1 \le i \le n\text{ },$$

*are disjoint.*

*Proof* By the Ring Lemma (Lemma 4.2) there exists a constant B < ∞ such that for any C > 1, any <sup>z</sup> satisfying <sup>z</sup> <sup>≥</sup> <sup>1</sup> <sup>−</sup> <sup>C</sup>−<sup>n</sup> and any 1 <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>n</sup>, if a circle of P intersects Az(C−<sup>i</sup> , 2C−<sup>i</sup> ) or is tangent to a circle that intersects Az(C−<sup>i</sup> , 2C−<sup>i</sup> ), then its radius is at most BC−<sup>i</sup> . Hence, this set of circles is contained in the disc of radius (<sup>2</sup> <sup>+</sup> <sup>4</sup>B)C−<sup>i</sup> around <sup>z</sup>. Furthermore, since <sup>|</sup>z| ≥ <sup>1</sup> <sup>−</sup> <sup>C</sup>−n, by the Ring Lemma again there exists b > 0 such that any such circle must be of distance at least bC−<sup>i</sup> from z. Hence, any fixed C > <sup>4</sup>+4<sup>B</sup> <sup>b</sup> satisfies the assertion of the lemma. 

**Lemma 7.12** *Let* <sup>z</sup> <sup>∈</sup> <sup>U</sup> *and* r > <sup>0</sup>*. Let* <sup>U</sup> *be a uniform random variable in* [1, 2] *and denote by* μr *the law of the random set* Vz(U r, U r) *(as defined in Definition 7.10). Then there exists a constant* C < ∞ *depending only on the maximal degree such that*

$$
\mathcal{E}(\mu\_r) \le \mathcal{C} \dots
$$

*Proof* For each vertex v, the event v ∈ Vz(U r, U r) implies that the circle {w ∈ <sup>C</sup> : |<sup>w</sup> <sup>−</sup> <sup>z</sup>| = U r} intersects the circle P (v) or intersects P (f ) for some face <sup>f</sup> incident to v. The union of P (v) and P (f ) over all such faces f is contained in the Euclidean ball around the center of P (v) of radius r(v) + 2 max<sup>f</sup> :v∈<sup>f</sup> r(f ). Since T has finite maximal degree we have that r(f ) ≤ Cr(v) for all f with v ∈ f where C < ∞ depends only on the maximal degree by the Ring Lemma (Lemma 4.2). Hence,

$$\mu\_r(\upsilon \in V\_\varepsilon(Ur, Ur)) \le \frac{1}{r} \min\left(2r(\upsilon) + 4 \max\_{f: f \ni v} r(f), \, r\right) \le \frac{C}{r} \min\{r(\upsilon), r\}.\tag{7.6}$$

We claim that

$$\sum\_{v \in V\_{\mathbb{C}}(r, 2r)} \min\{r(v), r\}^2 \le 16r^2. \tag{7.7}$$

Indeed, consider a vertex v ∈ Vz(r, 2r) for which the corresponding circle P (v) has radius larger than <sup>r</sup>. By Definition 7.10 this circle must intersect {<sup>w</sup> <sup>∈</sup> <sup>C</sup> : |w−z| ≤ 2r}. We replace each such P (v) with a circle of radius r that is contained in the original circle and intersects {<sup>w</sup> <sup>∈</sup> <sup>C</sup> : |<sup>w</sup> <sup>−</sup>z| ≤ <sup>2</sup>r}. The circles in this new set still have disjoint interiors and are contained in {<sup>w</sup> <sup>∈</sup> <sup>C</sup> : |<sup>w</sup> <sup>−</sup> <sup>z</sup>| ≤ <sup>4</sup>r}. Therefore their area is at most π16r<sup>2</sup> and (7.7) follows. The proof of lemma is now concluded by combining (7.6) and (7.7). 

*Proof of Theorem 7.1* Let F be a sample of WUSF<sup>G</sup> and given an edge e = (x, y) we define *A* <sup>e</sup> to be the event that x and y are in two distinct infinite connected components of F \ {e}. It is clear that every component of F is one-ended almost surely if and only if

$$\mathbf{P}(e \in \mathfrak{F}, \ \mathcal{A}'^{e}) = 0 \tag{7.8}$$

for every edge e of G. Consider the triangulation T described above Definition 7.10 and its circle packing P in U. By choosing the proper Möbius transformation we may assume that the tangency point between P (x) and P (y) is the origin, and that the centers of P (x) and P (y) lie on the negative and positive real axis, respectively.

Fix now an arbitrary ε > 0 and let Vε be all the vertices of G such that the center z(v) of P (v) satisfies <sup>|</sup>z(v)| ≤ <sup>1</sup> <sup>−</sup> <sup>ε</sup>. Denote by *<sup>B</sup>*<sup>e</sup> <sup>ε</sup> the event that every connected

**Fig. 7.1** Illustration of the proof. Left: On the event *A* <sup>e</sup> <sup>ε</sup> , the paths η<sup>x</sup> and η<sup>y</sup> split Vε into two pieces, *<sup>L</sup>* and *<sup>R</sup>*. Right: We define a random set containing a path (solid blue) from <sup>η</sup><sup>x</sup> to <sup>η</sup><sup>y</sup> ∪ {∞} in G \ Kc using a random circle (dashed blue). Here we see two examples, one in which the path ends at η<sup>y</sup> , and the other in which the path ends at the boundary (i.e., at infinity)

component of <sup>F</sup>\{e} intersects <sup>V</sup> \Vε. Note that *<sup>A</sup>*<sup>e</sup> ⊂ ∩ε>0*B*<sup>e</sup> <sup>ε</sup> but this containment is strict since it is possible that e ∈ F and x is connected to y in F inside Vε.

Assume that *B*<sup>e</sup> <sup>ε</sup> holds. Let <sup>η</sup><sup>x</sup> be the rightmost path in <sup>F</sup> \ {e} from <sup>x</sup> to <sup>V</sup> \ Vε when looking at <sup>x</sup> from <sup>y</sup>, and let <sup>η</sup><sup>y</sup> be the leftmost path in <sup>F</sup> \ {e} from <sup>y</sup> to V \ Vε when looking at y from x. As mentioned above, the paths ηx and ηy are not necessarily disjoint. Nonetheless, concatenating the reversal of η<sup>x</sup> with e and <sup>η</sup><sup>y</sup> separates Vε into two sets of vertices, *<sup>L</sup>* and *<sup>R</sup>*, which are to the left and right of e (when viewed from x to y) respectively. See Fig. 7.1 for an illustration of the case when ηx and ηy are disjoint (when they are not, *R* is a "bubble" separated from V \ Vε).

On the event *B*<sup>e</sup> <sup>ε</sup> , let K be the set of edges that are either incident to a vertex in *L* or belong to the path ηx ∪ηy , and set K = E off of this event. Note that the edges of <sup>K</sup> do not touch the vertices of *<sup>R</sup>*. The condition that <sup>η</sup><sup>x</sup> and <sup>η</sup><sup>y</sup> are the rightmost and leftmost paths to V \ Vε from x and y is equivalent to the condition that K does not contain any open path from <sup>x</sup> to <sup>V</sup> \ Vε other than <sup>η</sup>x, and does not contain any open path from <sup>y</sup> to <sup>V</sup> \ Vε other than <sup>η</sup>y. We note that <sup>K</sup> can be explored algorithmically, without querying the status of any edge in E \ K, by performing a right-directed depth-first search of x's component in F and a left-directed depth-first search of y's component in F, stopping each search when it first leaves Vε.

Denote by *A* <sup>e</sup> <sup>ε</sup> the event that ηx and ηy are disjoint, or equivalently, that K does not contain an open path from x to y (and in particular, no path starting at ηx and ending at ηy ). The event *A* <sup>e</sup> <sup>ε</sup> is measurable with respect to the random set K and *<sup>A</sup>* <sup>e</sup> = ∩ε>0*<sup>A</sup>* <sup>e</sup> <sup>ε</sup> . Hence

$$\mathbf{P}(e \in \mathfrak{F}, \ \omega \ell^{\ell}) \le \mathbf{P}(e \in \mathfrak{F} \mid \omega \ell^{\ell}\_{\varepsilon}) = \mathbf{E}[\mathbf{P}(e \in \mathfrak{F} \mid \omega \ell^{\ell}\_{\varepsilon}, K)].\tag{7.9}$$

Denote by Ko the open edge of K (that is, the edge of K in F) and by Kc the closed edges of K (that is, the edges of K not belonging to F). In particular, ηx and ηy are contained in Ko. Then by the UST Markov property (7.3), conditioned on K and the event *A* <sup>e</sup> <sup>ε</sup> , the law of F is equal to the union of Ko with a sample of the WUSF on (G − Kc)/Ko. In particular, by Kirchhoff's formula Theorem 7.5 we have that

$$\mathbf{P}(e \in \mathfrak{F} \mid \omega\_{\varepsilon}^{\prime e}, \mathbf{K}) \le \mathcal{R}\_{\text{eff}}^{W}(\eta\_{\chi} \leftrightarrow \eta\_{\chi}; \; G - \mathbf{K}\_{c})\,,\tag{7.10}$$

where in the last inequality we used the fact that gluing cannot increase the resistance (Corollary 2.30).

We will show that the last quantity tends to 0 as ε → 0 which gives (7.8). To that aim, let v<sup>x</sup> be the endpoint of the path η<sup>x</sup> and let z<sup>0</sup> be the center of the P (vx ). On the event *A* <sup>e</sup> <sup>ε</sup> , for each 1 − |z0| ≤ r ≤ 1/4, we claim that the set Vz<sup>0</sup> (r, r), as defined in Definition 7.10, contains a path in G from η<sup>x</sup> to η<sup>y</sup> that is contained in *<sup>R</sup>* <sup>∪</sup> <sup>η</sup><sup>x</sup> <sup>∪</sup> <sup>η</sup><sup>y</sup> or an infinite simple path starting at ηx that is contained in *<sup>R</sup>* <sup>∪</sup> <sup>η</sup>x. Either of these paths are therefore a path in G − Kc.

To see this, consider the arc A (z0,r) = {<sup>z</sup> <sup>∈</sup> <sup>U</sup> : |<sup>z</sup> <sup>−</sup> <sup>z</sup>0| = <sup>r</sup>} viewed in the clockwise direction and let A(z0,r) be the subarc beginning at the last intersection of A (z0,r) with a circle corresponding to a vertex in the trace of ηx, and ending at the first intersection after this time of A (z0,r) with either ∂U or a circle corresponding to a vertex in the trace of η<sup>y</sup> (see Fig. 7.1). Hence, if *A* <sup>e</sup> <sup>ε</sup> holds, then the set of vertices of T whose circles in P intersect A(z0,r) contains a path in <sup>T</sup> starting at <sup>η</sup><sup>x</sup> and ending <sup>η</sup><sup>y</sup> or does not end at all, for every 1 − |z0| ≤ <sup>r</sup> <sup>≤</sup> <sup>1</sup>/4. To obtain a path in G rather than T we divert the path counterclockwise around each face of G. That is, whenever the path passes from a vertex u of G to a face f of G and then to a vertex v of G, we replace this section of the path with the list of vertices of G incident to f that are between u and v in the counterclockwise order. By Definition 7.10 this diverted path is in Vz<sup>0</sup> (r, r) and so this construction shows that the subgraph of <sup>G</sup> <sup>−</sup> Kc induced by the set Vz<sup>0</sup> (r, r) contains a path from <sup>η</sup><sup>x</sup> to η<sup>y</sup> or an infinite path from ηx, as claimed.

Let ri <sup>=</sup> <sup>C</sup>−<sup>i</sup> for <sup>i</sup> <sup>=</sup> <sup>1</sup>,...,N where C < <sup>∞</sup> the constant from Lemma 7.11 and N = &logC(ε)'. Assume without loss of generality that C ≥ 4 so that ε ≤ ri ≤ 1/4 for all i = 1,...,N. By Lemma 7.11 the measures μri defined in Lemma 7.12 are supported on sets that are contained in the disjoint sets Vz(ri, 2ri). Thus, by Lemma 7.9 and Lemma 7.12 we have

$$\mathcal{R}\_{\text{eff}}^{W}\Big(\eta^{\chi}\leftrightarrow\eta^{\chi}\cup\{\infty\};\ G\backslash K\_{c}\Big)\leq\mathcal{E}\left(\frac{1}{N}\sum\_{l=1}^{N}\mu\_{r\_{l}}\right)=\frac{1}{N^{2}}\sum\_{l=1}^{N}\mathcal{E}(\mu\_{r\_{l}})\leq\frac{B}{\log(1/\varepsilon)},$$

where B < ∞ is a constant depending only on the maximum degree. By symmetry we also have

$$\mathcal{R}\_{\text{eff}}^{W}(\eta^{\text{y}} \leftrightarrow \eta^{\text{x}} \cup \{\infty\}; \ G - K\_c) \le \frac{B}{\log(1/\varepsilon)}.$$

Applying Lemma 7.8 and (7.10) gives

$$\mathbf{P}(e \in \mathfrak{F} \mid \mathcal{F}\_K, \ \mathcal{A}\_\varepsilon^{\ell}) \le \frac{\mathfrak{I}B}{\log(1/\varepsilon)}\,\,\mathrm{d}$$

We plug this estimate into (7.10) and take ε → 0, which together with (7.9) shows that (7.8) holds, concluding our proof. 

#### **7.6 Exercises**


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# **Chapter 8 Related Topics**

In this chapter we briefly review some aspects of the literature on circle packing that unfortunately we do not have space to get into in depth in this course. We hope this will be useful as a guide to further reading.

1. **Double circle packing.** If one wishes to study planar graphs that are *not* triangulations, it is often convenient to work with *double circle packings*, which enjoy similar rigidity properties to usual circle packings, but for the larger class of **polyhedral** planar graphs. Here, a planar graph is polyhedral if it is both simple and 3**-connected**, meaning that the removal of any two vertices cannot disconnect the graph. Double circle packings also satisfy a version of the ring lemma [45, Theorem 4.1], which means that they can be used to produce good straightline embeddings of polyhedral planar graphs that have bounded face degrees but which are not necessarily triangulations.

Let G be a planar graph with vertex set V and face set F. A double circle packing of <sup>G</sup> is a pair of circle packings <sup>P</sup> = {Pv : <sup>v</sup> <sup>∈</sup> <sup>V</sup> } and <sup>P</sup>† = {Pf : <sup>f</sup> <sup>∈</sup> F} satisfying the following conditions:


See Fig. 8.1 for an illustration.

Thurston's proof of the circle packing theorem also implies that every finite polyhedral planar graph admits a double circle packing. This was also shown by Brightwell and Scheinerman [13]. As with circle packings of triangulations, the double circle packing of any finite polyhedral planar map is unique up to

A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes in Mathematics 2243, https://doi.org/10.1007/978-3-030-27968-4\_8

#### 106 8 Related Topics

**Fig. 8.1** A finite polyhedral plane graph (left) and its double circle packing (right). Primal circles are filled and have solid boundaries, dual circles have dashed boundaries

Möbius transformations or reflections. The theory of double circle packings in the infinite setting follows from the work of He [37], and is exactly analogous to the corresponding theory for triangulations. Indeed, essentially everything we have to say in these notes about circle packings of simple triangulations can be generalized to double circle packings of polyhedral planar maps (sometimes under the additional assumption that the faces are of bounded degree).

	- If u, v ∈ V \ {∂} are distinct, then the closure of hvDv and huDu have disjoint interiors, and intersect if and only if v and u are adjacent in T .
	- If <sup>v</sup> <sup>∈</sup> <sup>V</sup> \ {∂}, then the closure of hvDv and <sup>C</sup> \ <sup>D</sup> have disjoint interiors, and intersect if and only if v is adjacent to ∂ in T .

In other words, we can represent the triangulation of T by a packing with arbitrary smooth convex shapes that are specified up to homothety (it is quite surprising at first that rotations are not needed). The full monster packing theorem also allows one to relax the smoothness and convexity assumptions above in various ways. The proof of the monster packing theorem is based upon Brouwer's fixed point theorem, and does not give an algorithm for computing the packing.

3. **Square tiling.** Another popular method of embedding planar graphs is the *square tiling*, in which vertices are represented by horizontal line segments and edges by squares; such square tilings can take place either in a rectangle, the plane, or a cylinder. Square tiling was introduced by Brooks et al. [14], and generalized to infinite planar graphs by Benjamini and Schramm [10]. Like circle packing,

square tiling can be thought of as a discrete version of conformal mapping, and in particular can be used to approximate the uniformizing map from a simply connected domain with four marked boundary points to a rectangle. For studying the random walk, a very nice feature of the square tiling that is not enjoyed by the circle packing is that the height of a vertex in the cylinder is a harmonic function, so that the height of a random walk is a martingale. Furthermore, Georgakopoulos [28] observed that if one stops the random walk at the first time it hits some height, then its horizontal coordinate at this time is uniform on the circle (this takes some interpretation to make precise). Further works on square tiling include [1, 28, 46] (Fig. 8.2).

Unlike circle packing, however, square tilings do not enjoy an analogue of the ring lemma, and can be geometrically very degenerate. Indeed, it is possible for edges to be represented by squares of zero area, and is also possible for two distinct planar graphs to have the same square tiling. Furthermore, square tilings are typically defined with reference to a specified root vertex, and it is difficult to compare the two different square tilings of the same graph that are computed with respect to different root vertices. These differences tend to mean that square tilings are best suited to quite different problems than circle packing.

We also remark that a different sort of square tiling in which *vertices* are represented by squares was introduced independently by Cannon et al. [15] and Schramm [74].

4. **Multiply-connected triangulations.** Several works have studied generalizations of the circle packing theorem to triangulations that are either not simply connected or not planar. Most notably, He and Schramm [39] proved that every triangulation of a domain with countably many boundary components can be circle packed in a circle domain, that is, a domain all of whose boundary components are either circles or points: see Fig. 8.3 for examples. The corresponding statement for a triangulation of an *arbitrary* domain is a major open problem, and is closely related to the Koebe conjecture.

Gurel-Gurevich, the current author, and Suoto [32] generalized the part of the He-Schramm Theorem concerning recurrence of the random walk as follows: A

**Fig. 8.3** Left: A circle packing in the multiply-connected circle domain <sup>U</sup> \ {0}. Right: A circle packing in a circle domain with several boundary components

bounded degree triangulation circle packed in a domain D is transient if and only if Brownian motion on D is transient, i.e. leaves D in finite time almost surely.

5. **Isoperimetry of planar graphs.** In [66], Miller, Teng, Thurston, and Vavasis used circle packing to give a new proof of the *Lipton-Tarjan planar separator theorem* [60], which concerns sparse cuts in planar graphs. Precisely, the theorem states that for any n-vertex planar graph, one can find a set of vertices of size at most O(√n) such that if this vertex set is deleted from the graph then every connected component that remains has size at most 3n/4. More precisely, the authors of [66] showed that if one circle packs a planar graph in the unit sphere of R3, normalizes by applying an appropriate Möbius transformation, and takes a random plane passing through the origin in R3, then the set of vertices whose corresponding discs intersect the plane will have the desired properties with high probability.

A related result of Jonasson and Schramm [47] concerns the *cover time* of planar graphs, i.e., the expected number of steps for a random walk on the graph to visit every vertex of the graph. They used circle packing to prove that the cover time of an n-vertex planar graph with maximum degree M is always at least cM n log2 n for some positive constant cM depending only on M. This bound is attained (up to the constant) for large boxes [−n, n] <sup>2</sup> in Z2. In general, it is possible for n-vertex graphs to have cover time as small as (1 + o(1))n log n.

6. **Boundary theory.** Benjamini and Schramm [9] proved that if P is a circle packing of a bounded degree triangulation in the unit disc U, then the simple random walk on the circle packed triangulation converges to a point in the boundary of U, and that the law of the limit point is non-atomic and has full support. (That is, the walk has probability zero of converging to any specific boundary point, and has positive probability of converging to any positive-length interval.) They used this result to deduce that a bounded degree planar graph admits non-constant bounded harmonic functions if and only if it is transient (equivalently, the invariant sigma-algebra of the random walk on the triangulation is non-trivial if and only if the walk is transient), and in this case it also admits non-constant bounded harmonic functions of finite Dirichlet energy. They also gave an alternative proof of the same result using square tiling instead of circle packing in [10].

Indeed, given the result of Benjamini and Schramm, one may construct a nonconstant bounded harmonic function h on T by taking any bounded, measurable function <sup>f</sup> : <sup>∂</sup><sup>U</sup> <sup>→</sup> <sup>R</sup> and defining <sup>h</sup> to be the *harmonic extension* of <sup>f</sup> , that is,

$$h(v) = \mathbf{E}\_v \left[ f \left( \lim\_{n \to \infty} z(X\_n) \right) \right],$$

where **E**<sup>v</sup> denotes expectation taken with respect to the random walk X started at v, and z(u) denotes the center of the circle in P corresponding to u. Angel et al. [6] proved that, in fact, *every* bounded harmonic function on a bounded degree triangulation can be represented in this way. In other words, the boundary ∂U can be identified with the **Poisson boundary** of the triangulation. Probabilistically, this means that the entire invariant σ-algebra of the random walk coincides with the σ-algebra generated by the limit point. They also proved the stronger result that ∂U can be identified with the **Martin boundary** of the triangulation. Roughly speaking, this means that every *positive* harmonic function on the triangulation admits a representation as the harmonic extension of some *measure* on ∂U. A related representation theorem for harmonic functions of *finite Dirichlet energy* on bounded degree triangulations was established by Hutchcroft [43].

The results of [6] regarding the Poisson boundary followed earlier work by Georgakopoulos [28], which established a corresponding result for square tilings. Both results were revisited in the work of Hutchcroft and Peres [46], which gave a simplified and unified proof that works for both embeddings.

A parallel boundary theory for circle packings of **unimodular random triangulations** of *unbounded* degree was developed by Angel, Hutchcroft, the current author, and Ray in [7].

7. **Harnack inequalities, Poincaré inequalities, and comparison to Brownian motion.** The work of Angel et al. [6] also established various quite strong estimates for random walk on circle packings of bounded degree triangulations. Roughly speaking, these estimates show that the random walk behaves similarly to the image of a Brownian motion under a *quasi-conformal map*, that is, a bijective map that distorts angles by at most a bounded amount (i.e., maps infinitesimal circles to infinitesimal ellipses of bounded eccentricity). These estimates were central to their result concerning the Martin boundary of the triangulation, and are also interesting in their own right. Further related estimates have also been established by Chelkak [17].

Recent work of Murugan [67] has built further upon these methods to establish very precise control of the random walk on (graphical approximations of) various deterministic self-similar fractal surfaces.

8. **Liouville quantum gravity and the KPZ correspondence.** Statistical physics in two dimensions has been one of the hottest areas of probability theory in recent years. The introduction of Schramm's SLE [75] and further breakthrough developments by Lawler, Schramm and Werner (see [53, 54] and the references within) on the one hand, and the application of discrete complex analysis, pioneered by Smirnov [79], on the other, have led to several breakthroughs and to the resolution of a number of long-standing conjectures. These include the conformally invariant scaling limits of critical percolation [77] and Ising models [78], and the determination of critical exponents and dimensions of sets associated with planar Brownian motion [53] (such as the frontier and the set of cut points). It is manifest that much progress will follow, possibly including the treatment of self-avoiding walk (the connective constant of the hexagonal lattice was calculated in the breakthrough work [22]), the O(n) loop model and the Potts model. While the bulk of this body of work applies to specific lattices, there are many fascinating problems in extending results to arbitrary planar graphs.

The next natural step is to study the classical models of statistical physics in the context of random planar maps (see Le Gall's 2014 ICM proceedings [57]). There are deep conjectured connections between the behaviour of the models in the random setting versus the Euclidean setting, most significantly the KPZ formula of Knizhnik et al. [50] from conformal field theory. This formula relates the dimensions of certain sets in Euclidean geometry to the dimensions of corresponding sets in the random geometry. It may provide a systematic way to analyze models on the two dimensional Euclidean lattice: first study the model in the random geometry setting, where the Markovian properties of the underlying space make the model tractable; then use the KPZ formula to translate the critical exponents from the random setting to the Euclidean one.

Much of this picture is conjectural but a definite step towards this goal was taken in the influential paper of Duplantier and Sheffield [23]. Let us describe their formulation. Let Gn be a random triangulation on n vertices and consider its circle packing (or any other "natural" embedding) in the unit sphere. The embedding induces a random measure μn on the sphere by putting μn(A) to be the proportion of circle centers that are in A. The Duplantier-Sheffield conjecture asserts that the measures μn converge in distribution to a random measure μ on the sphere that has density given by an exponential of the Gaussian free field the latter is carefully defined and constructed in [23]. This measure is what is known as *Liouville quantum gravity* (LQG).

Next, given a deterministic or random set K on the sphere, one can calculate its expected dimension using the random measure given by LQG, and using the usual Lebesgue measure—one gets two different numbers. Duplantier and Sheffield [23] obtain a quadratic formula allowing to compute one number from the other in the spirit of [50]; this is the first rigorous instance of the KPZ correspondence. It allows one to compute the dimension of random sets in the Z<sup>2</sup> lattice (corresponding to Lebesgue measure) by first calculating the corresponding dimension in the random geometry setting and then appealing to the KPZ formula.

Many difficult models of statistical physics are tractable on a random planar map due to the inherent randomness of the space. For instance, it can be shown that the self avoiding walk on the UIPT behaves diffusively, that is, the endpoint of a self avoiding walk of length n is typically of distance n1/2+o(1) from the origin [19, 34]. A straightforward calculation with the KPZ formula allows one to predict that the typical displacement of the self-avoiding walk of length n on the lattice Z<sup>2</sup> is n3/4+o(1) —a notoriously hard open problem with endless simulations supporting it.

LQG and the KPZ correspondence thus pose a path to solving many difficult problems in classical two-dimensional statistical physics. We refer the interested reader to Garban's excellent survey [27] of the topic.

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## **References**


© The Author(s) 2020

A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes

in Mathematics 2243, https://doi.org/10.1007/978-3-030-27968-4


## **Index**

antisymmetry, 14

balanced rooted tree, 73

canopy tree, 63 carrier, 3, 50 circle packing theorem, 1, 34 commute time identity, 21 conductance, 11 convergence in distribution, 62 current flow, 14 cutset, 27

degree biasing and unbiasing, 78 Dirichlet's principle, 24 disk triangulation, 47 double circle packing, 105 drawing, 33 dual graph, 95

effective conductance, 18 effective resistance, 18 effective resistance in an infinite network, 26, 97 energy of a flow, 23 energy of a function, 24 essential spanning forests, 93 Euler's formula, 34 exhaustive sequence, 25

face, 34 flow, 14 flow from a to ∞, 26 free effective resistance, 94 free uniform spanning forest, 90, 93

gluing, 20

harmonic function, 11 He-Schramm's theorem, 50 hitting time, 13

infinite planar map, 47 inner face, 34 isolation radius, 64

Kirchhoff's cycle law, 15 Kirchhoff's effective resistance formula, 91 Kirchhoff's node law, 14

local convergence, 62 local limit, 62 locally finite, 25, 47

magic lemma, 64 Markings, 80 Martin boundary, 109 method of random paths, 29 method of random sets, 98

© The Author(s) 2020 A. Nachmias, *Planar Maps, Random Walks and Circle Packing*, Lecture Notes in Mathematics 2243, https://doi.org/10.1007/978-3-030-27968-4

one-ended, 3, 48 outer face, 34

parallel law, 19 planar graph, 33 planar map, 33 planar separator theorem, 108 plane triangulation, 47 Poisson boundary, 109 polyhedral graph, 105 proper drawing, 33

Rayleigh's Monotonicity Law, 24 recurrent, 21 recurrent network, 26 resistance, 11 Ring Lemma, 49

series law, 19 simple random walk, 13 spanning tree, 89 spatial Markov property, 92 spherically symmetric tree, 20 square tiling, 106 star-tree transform, 73 stationary local limit, 77 stationary random graph, 77 strength of a flow, 16

Thomson's principle, 23 transient, 21 canopy tree, 64 network, 26 triangulation, 3, 35, 47

uniform spanning tree, 89 unit current flow, 17

voltage, 12

weighted random walk, 13 wired effective resistance, 94 wired finite exhaustion, 93 wired uniform spanning forest, 93 *LECTURE NOTES IN MATHEMATICS*

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