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

被引:3
作者
Alt, Johannes [1 ,2 ]
Ducatez, Raphael [3 ]
Knowles, Antti [1 ]
机构
[1] Univ Geneva, Sect Math, Geneva, Switzerland
[2] NYU, Courant Inst Math Sci, New York, NY 10012 USA
[3] ENS Lyon, Unite Math Pures & Appl UMPA, Lyon, France
基金
欧洲研究理事会; 瑞士国家科学基金会;
关键词
Random graph; random matrix; Poisson statistics; eigenvector localization; EXTREMAL EIGENVALUES; RANDOM MATRICES; LARGE DISORDER; UNIVERSALITY; DELOCALIZATION; EIGENVECTORS; DIFFUSION; ABSENCE;
D O I
10.1214/22-AOP1596
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
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.
引用
收藏
页码:277 / 358
页数:82
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