Critical random walk in random environment on trees
Let \(\Gamma\) be a tree without leaves, i.e., an infinite, locally finite, rooted acyclic graph with no vertices of degree one. An environment for random walk on \(\Gamma\) is a choice of transition probabilities \(q(\sigma, \tau)\) on the vertices of \(\Gamma\), non-zero iff \(\sigma\) and \(\tau\) are neighbors. When these transition probabilities are taken as random variables, the resulting process is called a random walk in a random environment (RWRE). If \(q(\sigma, \tau)\) is seen as the random resistance of the edge from \(\sigma\) to \(\tau\), the model may be interpreted as a random electrical network. The basic assumption is that the logarithms of the ratios of resistances of neighboring edges are i.i.d. with mean 0 and finite variance. From the summary: ``Then the resulting RWRE is transient if simple random walk on \(\Gamma\) is transient, but not vice versa. We obtain general transience criteria for such walks, which are sharp for symmetric trees of polynomial growth. In order to prove these criteria, we establish results on boundary crossing by tree-indexed random walks. These results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities of ordinary mean-zero finite variance random walks in one dimension, which are of independent interest.'' The paper builds on, and adds to, an important body of earlier work by the authors and their coworkers. [Recent articles are, e.g., \textit{I. Benjamini} and the second author, Probab. Theory Relat. Fields 98, No. 1, 91-112 (1994; Zbl 0794.60068); \textit{R. Lyons}, Ann. Probab. 20, No. 4, 2043-2088 (1992; Zbl 0766.60091); \textit{R. Lyons} and the first author, ibid. 20, No. 1, 125-136 (1992; Zbl 0751.60066); the first author, in: Seminar on stochastic processes, 1992. Prog. Probab. 33, 221-239 (1992; Zbl 0786.60118)].
scientific article; zbMATH DE number 796535