Fluctuations Of Linear Spectral Statistics Of Deformed Wigner Matrices

Abstract

We investigate the fluctuations of linear spectral statistics of a Wigner matrix $W\_N$ deformed by a deterministic diagonal perturbation $D\_N$, around a deterministic equivalent which can be expressed in terms of the free convolution between a semicircular distribution and the empirical spectral measure of $D\_N$. We obtain Gaussian fluctuations for test functions in $\mathcal{C}\_c^7(\mathbb{R})$ ($\mathcal{C}\_c^2(\mathbb{R})$ for fluctuations around the mean). Furthermore, we provide as a tool a general method inspired from Shcherbina and Johansson to extend the convergence of the bias if there is a bound on the bias of the trace of the resolvent of a random matrix. Finally, we state and prove an asymptotic infinitesimal freeness result for independent GUE matrices together with a family of deterministic matrices, generalizing the main result from [Shl18].

References in corpus (5)

• 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
• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants
• Weak detection in the spiked Wigner model

Cited by in corpus (3)

• Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices
• Quantitative CLT for linear eigenvalue statistics of Wigner matrices
• Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices