Related papers: Eigenvector Statistics of L\'{e}vy Matrices

We consider inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency matrix and its…

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance $0\leq \sigma^2\leq 1$. In this paper, we present an alternative proof of this result for the edge…

In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices $XX^*$ with isotropic log-concave $X$-columns. A main example is the covariance estimator of the uniform measure on isotropic convex…

In the past 20 years, the study of real eigenvalues of non-symmetric real random matrices has seen important progress. Notwithstanding, central questions still remain open, such as the characterization of their asymptotic statistics and the…

In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm…

As in random matrix theories, eigenvector/value distributions are important quantities of random tensors in their applications. Recently, real eigenvector/value distributions of Gaussian random tensors have been explicitly computed by…

We propose to quantify the complexity of non-equilibrium steady state density operators, as well as of long-lived Liouvillian decay modes, in terms of level spacing distribution of their spectra. Based on extensive numerical studies in a…

Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments…

We prove estimates for $\mathbb{E} \| X: \ell_{p'}^n \to \ell_q^m\|$ for $p,q\ge 2$ and any random matrix $X$ having the entries of the form $a_{ij}Y_{ij}$, where $Y=(Y_{ij})_{1\le i\le m, 1\le j\le n}$ has i.i.d. isotropic log-concave…

Heavy-tailed distributions naturally occur in many real life problems. Unfortunately, it is typically not possible to compute inference in closed-form in graphical models which involve such heavy-tailed distributions. In this work, we…

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth…

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in…

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts,…

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the…

We consider the ensemble of real symmetric random matrices $H^{(n,\rho)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $\rho/n$. We prove that the normalized…

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example,…

The density of complex eigenvalues of random asymmetric $N\times N$ matrices is found in the large-$N$ limit. The matrices are of the form $H_0+A$ where $A$ is a matrix of $N^2$ independent, identically distributed random variables with…

We extend to the matrix setting a recent result of Srivastava-Vershynin about estimating the covariance matrix of a random vector. The result can be in- terpreted as a quantified version of the law of large numbers for positive…

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an…

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices $\frac1N(D_n\circ X_n)(D_n\circ X_n)^*$, studied in Girko 2001. Here, $X_n=(x_{ij})$ is an $n\times N$ random matrix consisting of independent…