A non-symmetric Kesten criterion and ratio limit theorem for random walks on amenable groups

TL;DR

This paper extends Kesten's criterion to non-symmetric random walks on amenable groups and establishes a ratio limit theorem, broadening understanding of spectral properties and asymptotic behavior in these groups.

Contribution

It provides a non-symmetric analogue of Kesten's spectral radius criterion and proves a ratio limit theorem for random walks on amenable groups.

Findings

Spectral radius equals one for non-symmetric walks on amenable groups.

Established a ratio limit theorem for these groups.

Extended Kesten's criterion beyond symmetric cases.

Abstract

We consider random walks on countable groups. A celebrated result of Kesten says that the spectral radius of a symmetric walk (whose support generates the group as a semigroup) is equal to one if and only if the group is amenable. We give an analogue of this result for finitely supported walks which are not symmetric. We also conclude a ratio limit theorem for amenable groups.