## Random walk in a random environment and first-passage percolation on trees.(English)Zbl 0751.60066

From the authors’ summary: 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. 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.

### MSC:

 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation 60F10 Large deviations
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