The measurable Kesten theorem

TL;DR

This paper establishes explicit relationships between spectral properties and cycle densities in finite d-regular graphs, proving new bounds on girth, characterizing Ramanujan graphs, and generalizing Kesten's theorem to unimodular random graphs.

Contribution

It provides explicit estimates linking spectral radius and cycle densities, proves girth bounds for Ramanujan graphs, and extends Kesten's theorem to unimodular random graphs.

Findings

Essential girth of Ramanujan graphs grows at least logarithmically double-logarithmically with graph size.

Infinite Ramanujan unimodular random graphs are trees.

Spectral measures of finite graphs approximate those of the infinite d-regular tree under certain conditions.

Abstract

We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. Kesten showed that if a Cayley graph has the same spectral radius as its universal cover, then it must be a tree. We generalize this to unimodular random graphs.