Harmonic maps on amenable groups and a diffusive lower bound for random walks

Details Harmonic maps on amenable groups and a diffusive lower bound for random walks Harmonic maps on amenable groups and a diffusive lower bound for random walks

2009-11-02

arxiv.org/abs/0911.0274v5

Mathematics

Probability

Scientific paper

We prove that on any infinite, connected, locally finite, transitive graph G, the probability of the random walk being within $\eps \sqrt{t}$ of the origin after t steps is at most $O(\eps)$. A similar statement holds for finite graphs, up to the relaxation time of the walk. Our approach uses non-constant equivariant harmonic mappings taking values in a Hilbert space. For the special case of discrete, amenable groups, we present a more explicit proof of the Mok-Korevaar-Schoen theorem on existence of such harmonic maps by constructing them from the heat flow on a Folner set.

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