The measurable Kesten theorem

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.

The previous, longer version 1 has been split in two parts: the present paper, and a more group-theoretic one with the title "Kesten's theorem for Invariant Random Subgroups"

References in corpus (2)

• Kesten's theorem for Invariant Random Subgroups
• Cycle density in infinite Ramanujan graphs

Cited by in corpus (6)

• Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics
• Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues
• Cycle density in infinite Ramanujan graphs
• Furstenberg Entropy of Intersectional Invariant Random Subgroups
• On the Local Geometry of Graphs in Terms of Their Spectra
• Approximate homomorphisms and sofic approximations of orbit equivalence relations