The Annals of Probability

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

James R. Lee and Yuval Peres

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We prove diffusive lower bounds on the rate of escape of the random walk on infinite transitive graphs. Similar estimates hold for finite graphs, up to the relaxation time of the walk. Our approach uses nonconstant 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 the existence of such harmonic maps by constructing them from the heat flow on a Følner set.

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Ann. Probab., Volume 41, Number 5 (2013), 3392-3419.

First available in Project Euclid: 12 September 2013

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Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60G42: Martingales with discrete parameter 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Random walks on groups rate of escape harmonic maps


Lee, James R.; Peres, Yuval. Harmonic maps on amenable groups and a diffusive lower bound for random walks. Ann. Probab. 41 (2013), no. 5, 3392--3419. doi:10.1214/12-AOP779.

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