The measurable kesten theorem

Abért, Miklós and Glasner, Yair and Virág, Bálint (2016) The measurable kesten theorem. Annals of Probability, 44 (3). pp. 1601-1646. ISSN 0091-1798

Preview

Text 1111.2080.pdf Download (641kB) | Preview

Abstract

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigidity result. 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. In particular, G contains few short cycles. In contrast, we show that d-regular unimodular random graphs with maximal growth are not necessarily trees. © Institute of Mathematical Statistics, 2016.

Actions (login required)

Edit Item