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=1n(aij)1≤i,j≤n∈Rn×n symmetric (aij)1≤i≤j≤n i.i.d. standard normal, fluctuations of its largest eigenvalue λ1(A) asymptotically described by a real-valued 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 tail a11 decays sufficiently fast: limx→∞x4P(|a11|>x)=0, whereas law is regularly varying index α∈(0,4), ca(n)n1/2−2/αλ1(A) converges Fréchet ca:(0,∞)→(0,∞), slowly depending solely on a11. This paper considers family edge cases, limx→∞x4P(|a11|>x)=c∈(0,∞), unveils new type limiting behavior λ1(A): continuous function which 2, almost sure limit light-tailed case, plays pivotal role: f(x)=2,0<x<1,x+1 x,x≥1.