The Annals of Probability
- Ann. Probab.
- Volume 20, Number 1 (1992), 125-136.
Random Walk in a Random Environment and First-Passage Percolation on Trees
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.
Ann. Probab., Volume 20, Number 1 (1992), 125-136.
First available in Project Euclid: 19 April 2007
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A43
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