Seminar in probability theory: Antti Knowles (Geneva)

Local law and eigenvector delocalization for supercritical Erdos-Renyi graphs

We consider the adjacency matrix of the Erdos-Renyi graph G(N,p) in the supercritical regime pN > C log N for some universal constant C. We show that the eigenvalue density is with high probability well approximated by the semicircle law on all spectral scales larger than the typical eigenvalue spacing. We also show that all eigenvectors are completely delocalized with high probability. Both results are optimal in the sense that they are known to be false for pN < log N. A key ingredient of the proof is a new family of large deviation estimates for multilinear forms of sparse vectors.

Joint work with Yukun He and Matteo Marcozzi.

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