Pemantle, Robin; Peres, Yuval Critical random walk in random environment on trees. (English) Zbl 0837.60066 Ann. Probab. 23, No. 1, 105-140 (1995). 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., I. Benjamini and the second author, Probab. Theory Relat. Fields 98, No. 1, 91-112 (1994; Zbl 0794.60068); R. Lyons, Ann. Probab. 20, No. 4, 2043-2088 (1992; Zbl 0766.60091); 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)]. Reviewer: G.Högnäs (Åbo) Cited in 15 Documents MSC: 60G50 Sums of independent random variables; random walks 60G60 Random fields 60G70 Extreme value theory; extremal stochastic processes 60E07 Infinitely divisible distributions; stable distributions Keywords:random electrical network; Hausdorff dimension; backward push; capacity; boundary crossing; random walk in a random environment; random resistance; comparison inequalities; percolation processes Citations:Zbl 0794.60068; Zbl 0766.60091; Zbl 0751.60066; Zbl 0786.60118 PDFBibTeX XMLCite \textit{R. Pemantle} and \textit{Y. Peres}, Ann. Probab. 23, No. 1, 105--140 (1995; Zbl 0837.60066) Full Text: DOI