Random matrices: localization of the eigenvalues and the necessity of four moments

Abstract. Consider the eigenvalues λi(Mn) (in increasing order) of a random Hermitian matrix Mn whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner’s semicircular law, one expects that λi(Mn) concentrates around γi n, where ∫ γi − ∞ ρsc(x)dx = i n and ρsc is the semicircular function. In this paper, we show that if the entries have vanishing third moment, then for all 1 ≤ i ≤ n E|λi(Mn)− nγi|2 = O(min(n−c min(i, n+ 1 − i)−2/3n2/3, n1/3+ε)), for some absolute constant c> 0 and any absolute constant ε> 0. In partic-ular, for the eigenvalues in the bulk (min{i, n − i} = Θ(n))