Results 1  10
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48
Uniform spanning forests
 Ann. Probab
, 2001
"... Abstract. 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 coincid ..."
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Cited by 61 (22 self)
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Abstract. 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 29 (6 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
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
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Hyperlinear and sofic groups: a brief guide
 Bull. Symbolic Logic
"... Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely re ..."
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Cited by 17 (1 self)
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Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless.
Sparse graphs: metrics and random models
"... Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the d ..."
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Cited by 9 (0 self)
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Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for largescale realworld networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasirandom graphs are in a sense completely
Identities and Inequalities for Tree Entropy
, 2007
"... Abstract. The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy and use one of them to prove that tree entropy respects stochastic ..."
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Cited by 6 (1 self)
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Abstract. The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy and use one of them to prove that tree entropy respects stochastic domination. We also prove that tree entropy is nonnegative in the unweighted case. §1. Introduction. The enumeration of spanning trees in a finite graph is a classical subject dating to the mid 19th century. Asymptotics began to play a role over 100 years later, in the 1960s. When a sequence of finite graphs converges in an appropriate, but very general, sense, Lyons (2005) gave a formula for the limit of the numbers of spanning trees in that sequence
Sparse random graphs with clustering
 In preparation
"... In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random ..."
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Cited by 5 (5 self)
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In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (nonPoisson) multitype branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with powerlaw degree sequences with a wide range of degree exponents and clustering coefficients. 1 Introduction and