EMERGENCE OF EXTENDED STATES AT ZERO IN THE SPECTRUM OF SPARSE RANDOM GRAPHS

• 成果类型: Article
• 署名作者: Coste, Simon; Salez, Justin
• 署名单位: Universite PSL; Universite Paris-Dauphine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1499
• 发表日期: 2021
• 页码: 2012-2030
• 关键词: absolutely continuous-spectrum; eigenvalue distribution

摘要

We confirm the long-standing prediction that c = e approximate to 2.718 is the thresh-old for the emergence of a nonvanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdos-Renyi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent (A - z)(-1) near the singular point z = 0, where A is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.