POISSON STATISTICS AND LOCALIZATION AT THE SPECTRAL EDGE OF SPARSE ERDOS-RENYI GRAPHS

• 成果类型: Article
• 署名作者: Alt, Johannes; Ducatez, Raphael; Knowles, Antti
• 署名单位: University of Geneva; New York University; Ecole Normale Superieure de Lyon (ENS de LYON)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1596
• 发表日期: 2023
• 页码: 277-358
• 关键词: extremal eigenvalues; RANDOM MATRICES; large disorder; UNIVERSALITY; delocalization; eigenvectors; diffusion; absence

摘要

We consider the adjacency matrix A of the Erdos-Renyi graph on N ver-tices with edge probability d/N. For (log log N)4 << d < log N, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson point process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale d kappa log N, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known dis-tribution. Together with (Comm. Math. Phys. 388 (2021) 507-579), our result establishes the coexistence of a fully delocalized phase and a fully localized phase in the spectrum of A. The proof relies on a three-scale rigidity argu-ment, which characterizes the fluctuations of the eigenvalues in terms of the fluctuations of sizes of spheres of radius 1 and 2 around vertices of large degree.