Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime

• 成果类型: Article; Early Access
• 署名作者: Han, Yi
• 署名单位: University of Cambridge; Massachusetts Institute of Technology (MIT)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01329-6
• 发表日期: 2024
• 关键词: sufficient condition; LARGEST EIGENVALUE; edge universality

摘要

Given An:=1n(a(ij)) an nxn symmetric random matrix, with elements above the diagonal given by i.i.d. random variables having mean zero and unit variance. It is known that when limx ->infinity x(4)P(|a(ij)|>x)=0, then fluctuation of the largest eigenvalue of An follows a Tracy-Widom distribution. When the law of aij is regularly varying with index alpha is an element of(0,4), then the largest eigenvalue has a Fr & eacute;chet distribution. An intermediate regime is recently uncovered in Diaconu (Ann Probab 51(2):774-804, 2023): when limx ->infinity x(4)P(|a(ij)|>x)=c is an element of(0,infinity), then the law of the largest eigenvalue converges to a deformed Fr & eacute;chet distribution. In this work we vastly extend the scope where the latter distribution may arise. We show that the same deformed Fr & eacute;chet distribution arises (1) for sparse Wigner matrices with an average of n(Omega)(1) nonzero entries on each row; (2) for periodically banded Wigner matrices with bandwidth p(n)=n(O(1)); and more generally for weighted adjacency matrices of any k(n)-regular graphs with k(n)=n(Omega(1)). In all these cases, we further prove that the joint distribution of the finitely many largest eigenvalues of An converge to a deformed Poisson process, and that eigenvectors of the outlying eigenvalues of An are localized, implying a mobility edge phenomenon at the spectral edge 2 for Wigner matrices. The sparser case with average degree n(o(1)) is also explored. Our technique extends to sample covariance matrices, proving for the first time that its largest eigenvalue still follows a deformed Fr & eacute;chet distribution, assuming the matrix entries satisfy lim(x ->infinity)x(4)P(|a(ij)|>x)=c is an element of(0,infinity). The proof utilizes a universality result recently established by Brailovskaya and Van Handel (Universality and sharp matrix concentration inequalities, 2022).