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38
Processes on unimodular random networks
 In preparation
, 2005
"... Abstract. 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 amen ..."
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Cited by 52 (4 self)
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Abstract. 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
Percolation perturbations in potential theory and random walks
, 1998
"... We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which ..."
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Cited by 31 (10 self)
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We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph a.s. admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, a.s. the infinite clusters of pBernoulli percolation also have nonconstant harmonic Dirichlet functions when p is sufficiently close to 1. Many conjectures and questions are presented.
Biased Random Walks on GaltonWatson Trees
 Fields
, 1996
"... . We consider random walks with a bias toward the root on the family tree T of a supercritical GaltonWatson branching process and show that the speed is positive whenever the walk is transient. The corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that ..."
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Cited by 27 (7 self)
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. We consider random walks with a bias toward the root on the family tree T of a supercritical GaltonWatson branching process and show that the speed is positive whenever the walk is transient. The corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that of the whole boundary. When the bias is directed away from the root and the extinction probability is positive, the speed may be zero even though the walk is transient; the critical bias for positive speed is determined. x1. Introduction. Consider a supercritical GaltonWatson branching process with generating function f(s) = P 1 k=0 p k s k , i.e., each individual has k offspring with probability p k , and m := f 0 (1) 2 (1; 1). Started with a single progenitor, this process yields a random infinite family tree T , called a GaltonWatson tree, on the event of nonextinction. We assume throughout that no p k is equal to 1. Simple random walk gives some information on the structure ...
GaltonWatson trees with the same mean have the same polar sets
 ANN. PROBAB
, 1995
"... Evans (1992) defines a notion of what it means for a set B to be polar for a process indexed by a tree,. The main result herein is that a tree picked from a GaltonWatson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a de ..."
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Cited by 17 (10 self)
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Evans (1992) defines a notion of what it means for a set B to be polar for a process indexed by a tree,. The main result herein is that a tree picked from a GaltonWatson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it also follows that certain random Cantor sets are capacityequivalent to each other and to deterministic Cantor sets.
Hausdorff Dimension Of The Harmonic Measure On Trees
 Systems
, 1997
"... . For a large class of Markov operators on trees we prove the formula HD = h=l connecting the Hausdorff dimension of the harmonic measure on the tree boundary, the rate of escape l and the asymptotic entropy h. Applications of this formula include random walks on free groups, conditional random w ..."
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Cited by 13 (5 self)
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. For a large class of Markov operators on trees we prove the formula HD = h=l connecting the Hausdorff dimension of the harmonic measure on the tree boundary, the rate of escape l and the asymptotic entropy h. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations. 0. Introduction The Hausdorff dimension HD¯ of a measure ¯ on a metric space (X; d) is defined as the minimal Hausdorff dimension of sets of full measure ¯ and shows the "degree of singularity" (or, of "fractalness" in the newspeak) of this measure. Even if the support of the measure ¯ is the whole space, HD¯ does not have to coincide with HDX. The Hausdorff dimension HD¯ characterizes the polynomial rate of decreasing of the measures ¯ of balls of the metric d around typical (with respect to ¯) points of X, in particular, if ball measures decrease regularly, i.e., the limit lim log ¯B(x; r)= log r = f...
Large deviations for random walks on GaltonWatson trees: averaging and uncertainty
 Prob. Th. Rel. Fields
, 2002
"... In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk fXng on a GaltonWa ..."
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Cited by 13 (6 self)
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In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk fXng on a GaltonWatson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by jXn j the distance between the node Xn and the root of T. Our main result is the almost sure equality of the large deviation rate function for jXn j=n under the "quenched measure" (conditional upon T), and the rate function for the same ratio under the "annealed measure" (averaging on T according to the GaltonWatson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a speci c tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when fXng is a biased walk on a GaltonWatson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a "ubiquity" lemma for GaltonWatson trees, due to Grimmett and Kesten (1984).
Equivalence relations with amenable leaves need not be amenable
, 1997
"... There are two notions of amenability for discrete equivalence relations. The "global" amenability (which is usually referred to just as "amenability") is the property of existence of leafwise invariant means, which, by a theorem of ConnesFeldmanWeiss, is equivalent to hyperfini ..."
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Cited by 9 (2 self)
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There are two notions of amenability for discrete equivalence relations. The "global" amenability (which is usually referred to just as "amenability") is the property of existence of leafwise invariant means, which, by a theorem of ConnesFeldmanWeiss, is equivalent to hyperfiniteness, or, to being the orbit equivalence relation of a Zaction. The notion of "local " amenability applies to equivalence relations endowed with an additional leafwise graph structure and means that a.e. leafwise graph is amenable (or, Følner) in the sense that it has subsets A with arbitrary small isoperimetric ratio j@Aj=jAj (equivalently, that 0 belongs to the spectrum of leafwise Laplacians). In the present article we exhibit examples showing that local amenability does not imply global amenability contrary to a widespread opinion expressed in a number of earlier papers. We construct these examples both in the measuretheoretical (for discrete equivalence relations) and in the smooth (for foliations of compact manifolds) categories. We also formulate a general criterion of global amenability in isoperimetric terms.
A Central Limit Theorem for biased random walks on GaltonWatson trees
, 2006
"... Let T be a rooted GaltonWatson tree with offspring distribution {pk} that has p0 = 0, mean m = ∑ kpk> 1 and exponential tails. Consider the λbiased random walk {Xn}n≥0 on T; this is the nearest neighbor random walk which, when at a vertex v with dv offspring, moves closer to the root with probabi ..."
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Cited by 9 (2 self)
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Let T be a rooted GaltonWatson tree with offspring distribution {pk} that has p0 = 0, mean m = ∑ kpk> 1 and exponential tails. Consider the λbiased random walk {Xn}n≥0 on T; this is the nearest neighbor random walk which, when at a vertex v with dv offspring, moves closer to the root with probability λ/(λ + dv), and moves to each of the offspring with probability 1/(λ+dv). It is known that this walk has an a.s. constant speed v = limn Xn/n (where Xn  is the distance of Xn from the root), with v> 0 for 0 < λ < m and v = 0 for λ ≥ m. For all λ ≤ m, we prove a quenched CLT for Xn  −nv. (For λ> m the walk is positive recurrent, and there is no CLT.) The most interesting case by far is λ = m, where the CLT has the following form: for almost every T, the ratio X[nt] / √ n converges in law as n → ∞ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of
Unsolved problems concerning random walks on trees
 Classical and Modern Branching Processes, K. Athreya and P. Jagers (editors
, 1997
"... Abstract. We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on GaltonWatson trees monotonic in the drift parameter? These random walks have been used in MonteCarl ..."
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Cited by 9 (1 self)
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Abstract. We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on GaltonWatson trees monotonic in the drift parameter? These random walks have been used in MonteCarlo algorithms for sampling from the vertices of a tree; in general, their behavior reflects the size and regularity of the underlying tree. Random walks are related to conductance. The distribution function for the conductance of GaltonWatson trees satisfies an interesting functional equation; is this distribution function absolutely continuous? §1. Introduction. To explore the structure of irregular trees, we consider nearestneighbor random walks on them. The behavior of simple random walk gives some information about the structure, but more can be gleaned by considering the oneparameter family of random walks RWλ described below. That is, the behavior of such random walks on spherically symmetric