The Annals of Probability

Random Walk in a Random Environment and First-Passage Percolation on Trees

Russell Lyons and Robin Pemantle

Full-text: Open access

Abstract

We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is likewise determined by the branching number of the tree. Finally, we show that the branching number determines the rate of first-passage percolation on trees, also known as the first-birth problem. Our techniques depend on quasi-Bernoulli percolation and large deviation results.

Article information

Source
Ann. Probab., Volume 20, Number 1 (1992), 125-136.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989920

Digital Object Identifier
doi:10.1214/aop/1176989920

Mathematical Reviews number (MathSciNet)
MR1143414

Zentralblatt MATH identifier
0751.60066

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A43

Keywords
Trees random walk random environment first-passage percolation first birth random networks

Citation

Lyons, Russell; Pemantle, Robin. Random Walk in a Random Environment and First-Passage Percolation on Trees. Ann. Probab. 20 (1992), no. 1, 125--136. doi:10.1214/aop/1176989920. https://projecteuclid.org/euclid.aop/1176989920


Export citation