On the Speed of Random Walks on Graphs
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چکیده
Lyons, Pemantle and Peres asked whether the asymptotic lower speed in an innnite tree is bounded by the asymptotic speed in the regular tree with the same average number of branches. In the more general setting of random walks on graphs, we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary, and prove this conjecture. We give sharp bounds for limiting speed (or, when applicable, sub-linear rate of escape) in terms of growth properties of the graph. For trees, we get a bound for the speed in terms of the Hausdorr dimension of the harmonic measure on the boundary. As a consequence, two conjectures of Lyons, Pemantle and Peres are resolved, and a new bound is given for the dimension of the harmonic measure deened by the biased random walk on a Galton-Watson tree.
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