Fluctuations of deformed Wigner random matrices

Abstract

Let Xn be a standard real symmetric (complex Hermitian) Wigner matrix, y1, y2, ..., yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn, α = n−1/2Xn + n−α/2(y1,..., yn), where 0 < α < 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn, α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation \(\mathbb{E}m_{n,\alpha } \left( z \right)\) and variance Var(mn, α(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation.