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 twodimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erd'os-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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 3679-3707 |
| Number of pages | 29 |
| Journal | Journal of the European Mathematical Society |
| Volume | 19 |
| Issue number | 12 |
| DOIs | |
| State | Published - 2017 |
Bibliographical note
Funding Information:Research of C.B. was partially supported by ANR-11-JS02-005-01. Research of A.S. was partially supported by DMS 1406247. Research of B.V. was partially supported by the Canada Research Chair program and the NSERC Discovery Accelerator Supplement.
Publisher Copyright:
© European Mathematical Society 2017.
Keywords
- Continuous spectra
- Erd'os-Rényi graph
- Expected spectral measure
- Sparse random graphs
- Supercritical percolation
- Unimodular tree