The measurable kesten theorem

Miklós Abért, Yair Glasner, Bálint Virág

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

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.

Original languageEnglish
Pages (from-to)1601-1646
Number of pages46
JournalAnnals of Probability
Volume44
Issue number3
DOIs
StatePublished - 1 Jan 2016

Keywords

  • Girth
  • Mass transport principal
  • Ramanujan graphs
  • Spectral radius
  • Unimodular random graphs

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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