Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices

Abstract

In this paper, we study the complex Wigner matrices Mn=1/√n Wn whose eigenvalues are typically in the interval [-2,2]. Let λ1≤λ2⋯≤λn be the ordered eigenvalues of Mn. Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [-2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as An[f; u]=∑l=1 nf(λl)1{λl≤u}. And the second one is Bn[f; k]=∑l=1 kf(λl) with positive integer k=kn such that k/n→y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from Bn[f; ⌊ nt⌋]. The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on An[f;u] for the real Wigner matrices will also be briefly discussed. © 2012 Springer Science+Business Media New York.