## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 1 (1995), 105-140.

### Critical Random Walk in Random Environment on Trees

Robin Pemantle and Yuval Peres

#### Abstract

We study the behavior of random walk in random environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree $\Gamma$ are i.i.d. random variables whose logarithms have mean zero and finite variance. 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 for ordinary mean-zero finite variance random walks in one dimension, which are of independent interest.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 1 (1995), 105-140.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988379

**Digital Object Identifier**

doi:10.1214/aop/1176988379

**Mathematical Reviews number (MathSciNet)**

MR1330763

**Zentralblatt MATH identifier**

0837.60066

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J15

Secondary: 60G60: Random fields 60G70: Extreme value theory; extremal processes 60E07: Infinitely divisible distributions; stable distributions

**Keywords**

Tree random walk random environment random electrical network tree-indexed process percolation boundary crossing capacity Hausdorff dimension

#### Citation

Pemantle, Robin; Peres, Yuval. Critical Random Walk in Random Environment on Trees. Ann. Probab. 23 (1995), no. 1, 105--140. doi:10.1214/aop/1176988379. https://projecteuclid.org/euclid.aop/1176988379