Results 1  10
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122
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 97 (25 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.
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 68 (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.
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 58 (7 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
A Critical Branching Process Model for Biodiversity
, 2008
"... Motivated as a null model for comparison with data, we study the following model for a phylogenetic tree on n extant species. The origin of the clade is a random time in the past, whose (improper) distribution is uniform on (0, ∞). After that origin, the process of extinctions and speciations is a c ..."
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Cited by 44 (5 self)
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Motivated as a null model for comparison with data, we study the following model for a phylogenetic tree on n extant species. The origin of the clade is a random time in the past, whose (improper) distribution is uniform on (0, ∞). After that origin, the process of extinctions and speciations is a continuoustime critical branching process of constant rate, conditioned on having the prescribed number n of species at the present time. We study various mathematical properties of this model as n → ∞ limits: time of origin and of most recent common ancestor; pattern of divergence times within lineage trees; time series of numbers of species; number of extinct species in total, or ancestral to extant species; and “local” structure of the tree itself. We emphasize several mathematical techniques: associating walks with trees, a point process representation of lineage trees, and Brownian limits.
On exchangeable random variables and the statistics of large graphs and hypergraphs
, 2008
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Borel oracles. an analytical approach to constanttime algorithms
 Proc. Amer. Math. Soc
"... In [9], Nguyen and Onak constructed the first constanttime algorithm for the approximation of the size of the maximum matching in bounded degree graphs. The Borel oracle machinery is a tool that can be used to convert some statements in Borel graph theory to theorems in the field of constanttime a ..."
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Cited by 27 (0 self)
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In [9], Nguyen and Onak constructed the first constanttime algorithm for the approximation of the size of the maximum matching in bounded degree graphs. The Borel oracle machinery is a tool that can be used to convert some statements in Borel graph theory to theorems in the field of constanttime algorithms. In this paper we illustrate the power of this tool to prove the existence of the above mentioned constanttime approximation algorithm.
Invariant transports of stationary random measures and massstationarity
, 2008
"... We introduce and study invariant (weighted) transportkernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu’s exchange formula. The second main r ..."
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Cited by 18 (6 self)
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We introduce and study invariant (weighted) transportkernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu’s exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transportkernels. We then introduce (in a nonstationary setting) the concept of massstationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is massstationary. 1. Introduction. We consider (jointly) stationary random measures on a locally compact Abelian group G, for instance, G = R d. A transportkernel is a Markovian kernel T that distributes mass over G and depends on both ω in the underlying sample space Ω and a location s ∈ G. The number