Harmonic maps on amenable groups and a diffusive lower bound for random walks
Probability
2013-10-04 v8 Metric Geometry
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 F{\o}lner set.
Cite
@article{arxiv.0911.0274,
title = {Harmonic maps on amenable groups and a diffusive lower bound for random walks},
author = {James R. Lee and Yuval Peres},
journal= {arXiv preprint arXiv:0911.0274},
year = {2013}
}
Comments
Published in at http://dx.doi.org/10.1214/12-AOP779 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)