A necessary and sufficient condition for edge universality of Wigner matrices

In this paper, we prove a necessary and sufficient condition for the Tracy–Widom law of Wigner matrices. Consider N×N symmetric Wigner matrices H with Hij=N−1/2xij whose upper-right entries xij (1≤i<j≤N) are independent and identically distributed (i.i.d.) random variables with distribution ν and diagonal entries xii (1≤i≤N) are i.i.d. random variables with distribution ν̃. The means of ν and ν̃ are zero, the variance of ν is 1, and the variance of ν̃ is finite. We prove that the Tracy–Widom law holds if and only if lim s→∞s4P(|x12|≥s)=0. The same criterion holds for Hermitian Wigner matrices.