The spectral edge of constant degree Erdős-Rényi graphs

Determining the spectrum and eigenvectors of the adjacency matrix of random graphs is a fundamental problem with applications in computer science and statistical physics. Often, the relevant model is the Erdős-Rényi model, where edges are included independently with some fixed probability. In this talk, we show that for Erdős-Rényi graphs with constant expected degree, the most positive and most negative eigenvalues and eigenvectors are completely localized, in that eigenvector entries decay away from individual, high-degree vertices, and eigenvalues are almost completely determined by the geometry surrounding these high-degree vertices. This answers a question of Alice Guionnet.

This talk is based on joint work with Ella Hiesmayer.