Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications

Abstract. 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 p-Bernoulli percolation also have nonconstant harmonic Dirichlet functions when p is sufficiently close to 1. Many conjectures and questions are presented. §1. Introduction. The question of whether various potential-theoretic properties of graphs and manifolds are preserved under perturbations or approximations has been studied for more than a

. We consider random walks with a bias toward the root on the family tree T of a supercritical Galton-Watson 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 Galton-Watson 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 Galton-Watson 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 ...

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 Galton-Watson 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 capacity-equivalent to each other and to deterministic Cantor sets.

. 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...

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 Galton-Watson 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 Galton-Watson 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 Galton-Watson 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 Galton-Watson trees, due to Grimmett and Kesten (1984).

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 Connes-Feldman-Weiss, is equivalent to hyperfiniteness, or, to being the orbit equivalence relation of a Z-action. 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 measure-theoretical (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.

Lyons, Pemantle and Peres asked whether the asymptotic lower speed in an infinite tree is bounded by the asymptotic speed in the regular tree with the same average number of branches. In the more general setting of random walks on graphs, we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary, and prove this conjecture. We give sharp bounds for limiting speed (or, when applicable, sub-linear rate of escape) in terms of growth properties of the graph. For trees, we get a bound for the speed in terms of the Hausdorff dimension of the harmonic measure on the boundary. As a consequence, two conjectures of Lyons, Pemantle and Peres are resolved, and a new bound is given for the dimension of the harmonic measure defined by the biased random walk on a Galton-Watson tree.

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 Galton-Watson trees monotonic in the drift parameter? These random walks have been used in Monte-Carlo 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 Galton-Watson trees satisfies an interesting functional equation; is this distribution function absolutely continuous? §1. Introduction. To explore the structure of irregular trees, we consider nearest-neighbor random walks on them. The behavior of simple random walk gives some information about the structure, but more can be gleaned by considering the one-parameter family of random walks RWλ described below. That is, the behavior of such random walks on spherically symmetric

Let S(v) be a function defined on the vertices v of the infinite binary tree. One algorithm to seek large positive values of S is the Metropolis-type Markov chain (Xn ) defined by P (Xn+1 = wjXn = v) = 1 3 e b(S(w)\GammaS(v)) 1 + e b(S(w)\GammaS(v)) for each neighbor w of v, where b is a parameter ("1=temperature") which the user can choose. We introduce and motivate study of this algorithm under a probability model for the objective function S, in which S is "tree-indexed simple random walk", that is the increments ¸(e) = S(w) \Gamma S(v) along parent-child edges e = (v; w) are independent and P (¸ = 1) = p; P (¸ = \Gamma1) = 1 \Gamma p. This algorithm has a "speed" r(p; b) = lim n n \Gamma1 ES(Xn ). We study the speed via a mixture of rigorous arguments, non-rigorous arguments and Monte Carlo simulations, and compare with a deterministic greedy algorithm which permits rigorous analysis. Formalizing the non-rigorous arguments presents a challenging problem. Mathematically, th...