In this paper, we prove a necessary and sufficient condition for the Tracy–Widom law of Wigner matrices. Consider symmetric Wigner matrices with whose upper-right entries are independent and identically distributed (i.i.d.) random variables with distribution and diagonal entries are i.i.d. random variables with distribution . The means of and are zero, the variance of is 1, and the variance of is finite. We prove that the Tracy–Widom law holds if and only if . The same criterion holds for Hermitian Wigner matrices.
Ji Oon Lee. Jun Yin. "A necessary and sufficient condition for edge universality of Wigner matrices." Duke Math. J. 163 (1) 117 - 173, 15 January 2014. https://doi.org/10.1215/00127094-2414767