Fluctuations of extreme eigenvalues of sparse Erdos-Renyi graphs

• 成果类型: Article
• 署名作者: He, Yukun; Knowles, Antti
• 署名单位: University of Zurich; University of Geneva
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01054-4
• 发表日期: 2021
• 页码: 985-1056
• 关键词: gaussian fluctuations; spectral statistics; RANDOM MATRICES; UNIVERSALITY; edge; LAW

摘要

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdos-Renyi graph G(N, p). We show that if N-epsilon <= Np <= N1/3-epsilon then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916-962, 2020) on the fluctuations of the extreme eigenvalues from Np >= N2/9+epsilon down to the optimal scale N-p >= N-epsilon. The main technical achievement of our proof is a rigidity bound of accuracy N-1/2-epsilon(Np)(-1/2) for the extreme eigenvalues, which avoids the (Np)(-1)-expansions from Erdos et al. (Ann Prob 41:2279-2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543-616, 2018). Our result is the last missing piece, added to Erdos et al. (Commun Math Phys 314:587-640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for N-p >= N-epsilon.