Results 1  10
of
69
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 ..."
Abstract

Cited by 136 (6 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 41 (15 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 20 (11 self)
 Add to MetaCart
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.
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 prob ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
(Show Context)
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
Large Deviations for Random Walks on GaltonWatson Trees: Averaging and Uncertainty
, 2001
"... ..."
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 ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
. 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...
Anchored expansion, percolation and speed
, 2004
"... Abstract. The anchored expansion of a graph G is the infimum over all finite connected vertex sets S that contain a fixed basepoint, of the boundarytovolume ratio ∂S/S. We solve several problems raised in a paper by Benjamini, Lyons and Schramm (1999) where this notion was introduced. We prove ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
Abstract. The anchored expansion of a graph G is the infimum over all finite connected vertex sets S that contain a fixed basepoint, of the boundarytovolume ratio ∂S/S. We solve several problems raised in a paper by Benjamini, Lyons and Schramm (1999) where this notion was introduced. We prove that the positivity of the anchored expansion is preserved under percolation with parameter p sufficiently close to 1, and also under a random stretch when the stretching law has an exponential tail. The importance of anchored expansion was exhibited by Virág (2000), who showed that positivity of anchored expansion implies that simple random walk on G has positive speed. We also study simple random walk in the infinite cluster of pBernoulli bond percolation in Cayley graphs of amenable groups of exponential growth known as “lamplighter groups”. We prove that at least for large p, the speed of random walk on the infinite cluster is positive if and only if the speed is positive for random walk on the whole group. §1. Introduction. Denote by V (G) and E(G), respectively, the sets of vertices and edges of an infinite