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

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

2004-04-02

arxiv.org/abs/math/0404045v1

Ann. Probab., 20, 125 - 136 (1992)

Mathematics

Probability

11 pages

Scientific paper

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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