Understanding the spectrum and eigenvectors of the adjacency matrix of random graphs is a fundamental problem with broad applications in computer science and statistical physics. A widely studied model is the Erdős-Rényi graph, where edges are included independently with a fixed probability. In this talk, we show that for Erdős-Rényi graphs with constant expected degree, the most positive and most negative eigenvalues 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 resolves a question of Alice Guionnet.

This talk is based on joint work with Ella Hiesmayer.