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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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991843

Digital Object Identifier
doi:10.1214/12-AOP779

Mathematical Reviews number (MathSciNet)
MR3127886

Zentralblatt MATH identifier
1284.05250

Subjects
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

Keywords
Random walks on groups rate of escape harmonic maps

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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