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

TL;DR

This paper establishes diffusive lower bounds for random walks on infinite transitive graphs, utilizing harmonic maps into Hilbert spaces, with explicit constructions for amenable groups based on heat flow.

Contribution

It provides a new proof for harmonic maps on amenable groups and derives lower bounds on the escape rate of random walks on various graphs.

Findings

Diffusive lower bounds for random walks on infinite transitive graphs.

Explicit harmonic map constructions for amenable groups.

Extension of bounds to finite graphs up to relaxation time.

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.