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September 2013 Harmonic maps on amenable groups and a diffusive lower bound for random walks
James R. Lee, Yuval Peres
Ann. Probab. 41(5): 3392-3419 (September 2013). DOI: 10.1214/12-AOP779


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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James R. Lee. Yuval Peres. "Harmonic maps on amenable groups and a diffusive lower bound for random walks." Ann. Probab. 41 (5) 3392 - 3419, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1284.05250
MathSciNet: MR3127886
Digital Object Identifier: 10.1214/12-AOP779

Primary: 20F65 , 60B15 , 60G42 , 60J45

Keywords: Harmonic Maps , random walks on groups , Rate of escape

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 5 • September 2013
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