MORE LIMITING DISTRIBUTIONS FOR EIGENVALUES OF WIGNER MATRICES

• 成果类型: Article
• 署名作者: Diaconu, Simona
• 署名单位: Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1614
• 发表日期: 2023
• 页码: 774-804
• 关键词: universality; edge; CONVERGENCE; spectrum

摘要

The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for A = 1/root n(a(ij))(1 <= i,j <= n) is an element of R-nxn symmetric with (a(ij))(1 <= i <= j <= n) i.i.d. standard normal, the fluctuations of its largest eigenvalue lambda(1)(A) are asymptotically described by a real-valued Tracy-Widom distribution TW1 : n(2/3)(lambda(1)(A) - 2) double right arrow TW1. As it often happens, Gaussianity can be relaxed, and this results holds when E[a(11)] = 0, E[a(11)(2)] = 1 and the tail of a(11) decays sufficiently fast: lim(x ->infinity) x(4)P(|a(11)| > x) = 0, whereas when the law of a(11) is regularly varying with index alpha is an element of (0, 4), c(a)(n)n(1/2-2/alpha) lambda(1)(A) converges to a Frechet distribution for c(a) : (0, infinity) -> (0, infinity), slowly varying and depending solely on the law of a(11). This paper considers a family of edge cases, lim(x ->infinity) x(4)P(|a(11)| > x) = c is an element of (0, infinity), and unveils a new type of limiting behavior for lambda(1)(A): a continuous function of a Frechet distribution in which 2, the almost sure limit of lambda(1)(A) in the light-tailed case, plays a pivotal role: f (x) = { 2, 0< x < 1, x+ 1 , x >= 1.