Kesten's theorem for uniformly recurrent subgroups

TL;DR

This paper extends Kesten's theorem to uniformly recurrent subgroups, providing bounds on spectral radii differences and showing that in Ramanujan graphs, random walks spend negligible time in short cycles.

Contribution

It introduces an inequality relating spectral radii of Cayley and Schreier graphs and extends Kesten's theorem to uniformly recurrent subgroups.

Findings

Spectral radius difference inequality established

Kesten's theorem extended to uniformly recurrent subgroups

Random walks in Ramanujan graphs spend o(n) time in short cycles

Abstract

We prove an inequality on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application we extend Kesten's theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs holds on average. More precisely, we show that if $\mathcal G$ is an infinite deterministic Ramanujan graph, then the time spent in short cycles by a random walk of length $n$ is $o(n)$.