The Annals of Probability

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

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 12 September 2013

https://projecteuclid.org/euclid.aop/1378991843

Digital Object Identifier
doi:10.1214/12-AOP779

Mathematical Reviews number (MathSciNet)
MR3127886

Zentralblatt MATH identifier
1284.05250

Citation

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. https://projecteuclid.org/euclid.aop/1378991843

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