Related papers: Linear Eigenvalue Statistics at the Cusp
The aim of this note is to provide a pedagogical survey of the recent works by the authors ( arXiv:1409.7548 and arXiv:1507.06013) concerning the local behavior of the eigenvalues of large complex correlated Wishart matrices at the edges…
We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix $\widetilde{W}$ and its minor $W$. We find that the fluctuation of this difference is much smaller than those of the…
Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$ iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of $X$ for…
We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…
We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough…
Consider a $N\times n$ matrix $\Sigma_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear…
For random matrices with block correlation structure we show that the fluctuations of linear eigenvalue statistics are Gaussian on all mesoscopic scales with universal variance which coincides with that of the Gaussian unitary or Gaussian…
A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under…
As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral…
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…
We compute exact asymptotic of the statistical density of random matrices belonging to invariant random matrices ensemble (RMT) orthogonal, unitary and symplectic ensembles, where all its eigenvalues lie within the interval $[\sigma,…
Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $\mu_1(M_n), \mu_2(M_n), ..., \mu_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$…
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often…
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…
The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of…
We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components.…
We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we…
A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non identically distributed. This phenomenon is at the heart of the asymptotic analysis…
The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little…
We study the eigenvector mass distribution of an $N\times N$ Wigner matrix on a set of coordinates $I$ satisfying $| I | \ge c N$ for some constant $c >0$. For eigenvectors corresponding to eigenvalues at the spectral edge, we show that the…