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 $\mathit{A}=\frac{1}{\sqrt{\mathit{n}}}{({\mathit{a}}_{\mathit{i}\mathit{j}})}_{1\le \mathit{i},\mathit{j}\le \mathit{n}}\in {\mathbb{R}}^{\mathit{n}\times \mathit{n}}$ symmetric with ${({\mathit{a}}_{\mathit{i}\mathit{j}})}_{1\le \mathit{i}\le \mathit{j}\le \mathit{n}}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue ${\mathit{\lambda }}_1(\mathit{A})$ are asymptotically described by a real-valued Tracy–Widom distribution ${\mathit{TW}}_1:{\mathit{n}}^{2/3}({\mathit{\lambda }}_1(\mathit{A})-2)\Rightarrow {\mathit{TW}}_1$. As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[{\mathit{a}}_{11}]=0$, $\mathbb{E}[{\mathit{a}}_{11}^2]=1$ and the tail of ${\mathit{a}}_{11}$ decays sufficiently fast: ${lim}_{\mathit{x}\to \infty }{\mathit{x}}^4\mathbb{P}(|{\mathit{a}}_{11}|>\mathit{x})=0$, whereas when the law of ${\mathit{a}}_{11}$ is regularly varying with index $\mathit{\alpha }\in (0,4)$, ${\mathit{c}}_{\mathit{a}}(\mathit{n}){\mathit{n}}^{1/2-2/\mathit{\alpha }}{\mathit{\lambda }}_1(\mathit{A})$ converges to a Fréchet distribution for ${\mathit{c}}_{\mathit{a}}:(0,\infty )\to (0,\infty )$, slowly varying and depending solely on the law of ${\mathit{a}}_{11}$. This paper considers a family of edge cases, ${lim}_{\mathit{x}\to \infty }{\mathit{x}}^4\mathbb{P}(|{\mathit{a}}_{11}|>\mathit{x})=\mathit{c}\in (0,\infty )$, and unveils a new type of limiting behavior for ${\mathit{\lambda }}_1(\mathit{A})$: a continuous function of a Fréchet distribution in which 2, the almost sure limit of ${\mathit{\lambda }}_1(\mathit{A})$ in the light-tailed case, plays a pivotal role: