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112
Recurrence of distributional limits of finite planar graphs
 Electron. J. Probab
, 2001
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Processes on unimodular random networks
, 2007
"... We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability fr ..."
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Cited by 124 (6 self)
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We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasitransitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications
Uniform spanning forests
 ANN. PROBAB
, 2001
"... We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d ..."
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Cited by 89 (23 self)
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We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d � 4. In the present work, we extend Pemantle’s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation, and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following: • The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant. • The tail σfields of the WSF and the FSF are trivial on any graph. • On any Cayley graph that is not a finite extension of Z, all component trees of the WSF have one end; this is new in Z d for d � 5. • On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. • The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space H d is analyzed. • A Cayley graph is amenable iff for all ɛ> 0, the union of the WSF and Bernoulli percolation with parameter ɛ is connected. • Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees. We also present numerous open problems and conjectures.
Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that ..."
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Cited by 56 (5 self)
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Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasitransitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph
Critical percolation on any nonamenable group has no infinite clusters
 Ann. Probab
, 1999
"... We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study of groupinvariant percolation. The goal here is to present a simpler selfc ..."
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Cited by 51 (10 self)
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We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study of groupinvariant percolation. The goal here is to present a simpler selfcontained proof that easily extends to quasitransitive graphs with a unimodular automorphism group. The key tool is a ‘‘masstransport’’ method, which is a technique of averaging in nonamenable settings. 1. Introduction. The
Percolation in the Hyperbolic Plane
, 2000
"... Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation ..."
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Cited by 47 (4 self)
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Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ (0, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the PoissonVoronoiBernoulli percolation process on the intensity of the underlying Poisson process.
Indistinguishability of percolation clusters
 Ann. Probab
, 1999
"... Abstract. We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long ..."
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Cited by 46 (14 self)
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Abstract. We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. longrange order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products, and inequalities for pu. §1. Introduction. Grimmett and Newman (1990) showed that if T is a regular tree of sufficiently high degree, then there are p ∈ (0, 1) such that Bernoulli(p) percolation on T × Z has infinitely many infinite components a.s. Benjamini and Schramm (1996) conjectured that the same is true for any Cayley graph of any finitely generated nonamenable group. (A finitely
Quasiisometries and rigidity of solvable groups
, 2005
"... Abstract. In this note, we announce the first results on quasiisometric rigidity of nonnilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasiis ..."
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Cited by 41 (4 self)
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Abstract. In this note, we announce the first results on quasiisometric rigidity of nonnilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasiisometric to R⋉R n where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasiisometries of the solvable Lie group. Our classification of self quasiisometries for R⋉R n proves a conjecture made by Farb and Mosher in [FM3]. Our techniques for studying quasiisometries extend to some other classes of groups and spaces. In particular, we characterize groups quasiisometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain DiestelLeader graphs are not quasiisometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW, Wo1]. We also prove that certain nonunimodular, nonhyperbolic solvable Lie groups are not quasiisometric to finitely generated groups. The results in this paper are contributions to Gromov’s program for classifying finitely generated groups up to quasiisometry [Gr2]. We introduce a new technique for studying quasiisometries, which we refer to as coarse differentiation.
Coarse differentiation of quasiisometries I: Spaces not quasiisometric to Cayley graphs
, 607
"... Abstract In this paper, we prove that certain spaces are not quasiisometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasiisometric to a Cayley graph ..."
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Cited by 39 (5 self)
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Abstract In this paper, we prove that certain spaces are not quasiisometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasiisometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in our 2007 article "Quasiisometries and rigidity of solvable groups." In particular, this paper contains many steps in the proofs of quasiisometric rigidity of lattices in Sol and of the quasiisometry classification of lamplighter groups. The proofs of those results are completed in "Coarse differentiation of quasiisometries II; Rigidity for lattices in Sol and Lamplighter groups." The method used here is based on the idea of coarse differentiation introduced in our 2007 article.