More Limiting Distributions for Eigenvalues of Wigner Matrices

The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij}{)}_{1\le i,j\le n}\in {\mathbb{R}}^{n\times n}$ symmetric with $(a_{ij}{)}_{1\le i\le j\le n}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue ${\lambda }_1(A)$ are asymptotically described by a real-valued Tracy-Widom distribution $T{W}_1:$ $n^{2\mathrm{/}3}({\lambda }_1(A)-2)\Rightarrow T{W}_1\mathrm{.}$ As it often happens, Gaussianity can be relaxed, and this results holds when $$\mathbb{E}$[a_{11}]=0, $\mathbb{E}$[a^2_{11}]=1,$andthetailof$a_{11}$decayssufficientlyfast:$\lim_{x \to \infty}{x^4$\mathbb{P}$(|a_{11}|>x)}=0,$whereaswhenthelawof$a_{11}$isregularlyvaryingwithindex$\alpha \in (0,4),$ $c_a(n)n^{1\mathrm{/}2-2\mathrm{/}\alpha }{\lambda }_1(A)$ converges to a Fr'echet distribution for $c_a:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ slowly varying and depending solely on the law of $a_{11}\mathrm{.}$ 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 ${\lambda }_1(A):$ a continuous function of a Fr'echet distribution in which $2,$ the almost sure limit of ${\lambda }_1(A)$ in the light-tailed case, plays a pivotal role: $f(x)=\begin{cases} 2, & 0<x<1 \newline x+\frac{1}{x}, & x \geq 1 \end{cases}.$