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Register a new userThe largest eigenvalue of rank one deformation of large Wigner matrices
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The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.
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In this paper, we consider the addition of two matrices in generic position, namely A + U BU * , where U is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices A and B, the law of the largest eigenvalue satisfies a large deviation principle, in the scale N, with an explicit rate function involving the limit of spherical integrals. We cover in particular all the cases when A and B have no outliers.
We establish a large deviation principle for the process of the largest eigenvalue of an Hermitian Brownian motion. By a contraction principle, we recover the LDP for the largest eigenvalue of a rank one deformation of the GUE.
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of $X_N$ for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on the characteristic function method in [47], local laws for the Green function of $X_N$ in [3, 46, 51] and analytic subordination properties of the free additive convolution [24, 41]. We also prove the analogous results for high-dimensional sample covariance matrices.
In this article, we establish a limiting distribution for eigenvalues of a class of auto-covariance matrices. The same distribution has been found in the literature for a regularized version of these auto-covariance matrices. The original non-regularized auto-covariance matrices are non invertible which introduce supplementary diffculties for the study of their eigenvalues through Girkos Hermitization scheme. The key result in this paper is a new polynomial lower bound for the least singular value of the resolvent matrices associated to a rank-defective quadratic function of a random matrix with independent and identically distributed entries. Another improvement in the paper is that the lag of the auto-covariance matrices can grow to infinity with the matrix dimension.
We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comparison method [27] using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.
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