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=1n(aij)1≤i,j≤n∈Rn×n symmetric with (aij)1≤i≤j≤n i.i.d. standard normal, the fluctuations of its largest eigenvalue λ1(A) are asymptotically described by a real-valued Tracy–Widom distribution TW1:n2/3(λ1(A)−2)⇒TW1. As it often happens, Gaussianity can be relaxed, and this results holds when E[a11]=0, E[a112]=1 and the tail of a11 decays sufficiently fast: limx→∞x4P(|a11|>x)=0, whereas when the law of a11 is regularly varying with index α∈(0,4), ca(n)n1/2−2/αλ1(A) converges to a Fréchet distribution for ca:(0,∞)→(0,∞), slowly varying and depending solely on the law of a11. This paper considers a family of edge cases, limx→∞x4P(|a11|>x)=c∈(0,∞), and unveils a new type of limiting behavior for λ1(A): a continuous function of a Fréchet distribution in which 2, the almost sure limit of λ1(A) in the light-tailed case, plays a pivotal role: f(x)=2,0<x<1,x+1 x,x≥1.