Mean quantum percolation

  • Charles Bordenave

    University of Toulouse, France
  • Arnab Sen

    University of Minnesota, Minneapolis, USA
  • Bálint Virág

    University of Toronto, Canada

Abstract

We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure of bond percolation in the two-dimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erdős–Rényi graph and for the spectral measure of a unimodular random tree with at least two ends. We give examples of random graphs with purely continuous spectrum.

Cite this article

Charles Bordenave, Arnab Sen, Bálint Virág, Mean quantum percolation. J. Eur. Math. Soc. 19 (2017), no. 12, pp. 3679–3707

DOI 10.4171/JEMS/750