Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices

Abstract

We consider large-dimensional Hermitian or symmetric random matrices of the form $W=M+\vartheta V$ where $M$ is a Wigner matrix and $V$ is a real diagonal matrix whose entries are independent of $M$. For a large class of diagonal matrices $V$, we prove that the fluctuations of linear spectral statistics of $W$ for $C_c^2$ test function can be decomposed into that of $M$ and of $V$, and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae for the means and variances of the limiting distributions.

63 pages

References in corpus (3)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices