A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin

BVRcIc photometric evolution and flickering during the 2010 outburst of the recurrent nova U Scorpii

CCD BVRcIc photometric observations of the 2010 outburst of the recurrent nova U Scorpii are presented. The light-curve has a smooth development characterized by t2(V)=1.8 and t3(V)=4.1 days, close to the t2(V)=2.2 and t3(V)=4.3 days of 1999 outburst. The plateau phase in 2010 has been brighter, lasting shorter and beginning earlier than in the 1999 outburst. Flickering, with an amplitude twice larger in $I_{\rm C}$ than in $B$ band, was absent on day +4.8 and +15.7, and present on day +11.8, with a time scale of about half an hour.Comment: published March 1

Eigenvalue variance bounds for covariance matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of random covariance matrices, both in the bulk and at the edge of the spectrum. In a preceding paper, the author established analogous results for Wigner matrices and stated the results for covariance matrices. They are proved in the present paper. Relying on the LUE example, which needs to be investigated first, the main bounds are extended to complex covariance matrices by means of the Tao, Vu and Wang Four Moment Theorem and recent localization results by Pillai and Yin. The case of real covariance matrices is obtained from interlacing formulas

On the Central Limit Theorem for the Eigenvalue Counting Function of Wigner and Covariance matrices

This note presents some central limit theorems for the eigenvalue counting function of Wigner matrices in the form of suitable translations of results by Gustavsson and O'Rourke on the limiting behavior of eigenvalues inside the bulk of the semicircle law for Gaussian matrices. The theorems are then extended to large families of Wigner matrices by the Tao and Vu Four Moment Theorem. Similar results are developed for covariance matrices

FLUCTUATIONS OF LINEAR SPECTRAL STATISTICS OF DEFORMED WIGNER MATRICES

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 $\mathcal{C}_c^7(\R)$ ($\mathcal{C}_c^2(\R)$ 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]