The measurable Kesten theorem

• Abért, Miklós ^[1] ; Glasner, Yair ^[2] ; Virág, Bálint ^[3]
  1. [1] Institute of Mathematics

     Institute of Mathematics

     Warszawa, Polonia

     
  2. [2] Ben-Gurion University of the Negev

     Ben-Gurion University of the Negev

     Israel

     
  3. [3] University of Toronto

     University of Toronto

     Canadá

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 1601-1646
• Idioma: inglés
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• Resumen

  • 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 cloglog|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.