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January, 1992 Random Walk in a Random Environment and First-Passage Percolation on Trees
Russell Lyons, Robin Pemantle
Ann. Probab. 20(1): 125-136 (January, 1992). DOI: 10.1214/aop/1176989920

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.

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Russell Lyons. Robin Pemantle. "Random Walk in a Random Environment and First-Passage Percolation on Trees." Ann. Probab. 20 (1) 125 - 136, January, 1992. https://doi.org/10.1214/aop/1176989920

Information

Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0751.60066
MathSciNet: MR1143414
Digital Object Identifier: 10.1214/aop/1176989920

Subjects:
Primary: 60J15
Secondary: 60K35 , 82A43

Keywords: first birth , First-passage percolation , random environment , random networks , Random walk , trees

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.20 • No. 1 • January, 1992
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