More Limiting Distributions for Eigenvalues of Wigner Matrices

Abstract

The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue $λ_1(A)$ are asymptotically described by a real-valued Tracy-Widom distribution $TW_1:$ $n^{2/3}(λ_1(A)-2) \Rightarrow TW_1.$ As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[a_{11}]=0, \mathbb{E}[a^2_{11}]=1,$ and the tail of $a_{11}$ decays sufficiently fast: $\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=0,$ whereas when the law of $a_{11}$ is regularly varying with index $α\in (0,4),$ $c_a(n)n^{1/2-2/α}λ_1(A)$ converges to a Fréchet distribution for $c_a:(0,\infty) \to (0,\infty)$ slowly varying and depending solely on the law of $a_{11}.$ This paper considers a family of edge cases, $\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=c \in (0,\infty),$ and unveils a new type of limiting behavior for $λ_1(A):$ a continuous function of a Fréchet distribution in which $2,$ the almost sure limit of $λ_1(A)$ in the light-tailed case, plays a pivotal role: $f(x)=\begin{cases} 2, & 0<x<1 \newline x+\frac{1}{x}, & x \geq 1 \end{cases}.$