Fluctuations of deformed Wigner random matrices

Abstract

Let X_n be a standard real symmetric (complex Hermitian) Wigner matrix, y₁, y₂, ..., y_n a sequence of independent real random variables independent of X_n. Consider the deformed Wigner matrix H_{n, α} = n^−1/2X_n + n^−α/2(y₁,..., y_n), where 0 < α < 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform m_{n, 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(m_{n, 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.

Keywords

Asymptotic expansion / deformed Wigner matrice / Gaussian fluctuation / linear statistics / Stein’s equation

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