Related papers: Extreme gaps between eigenvalues of random matrice…

For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic…

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…

We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$, in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices, $A_N$. The most attention is paid to the distribution and typical fluctuations of…

We numerically study the extreme-value statistics of the Schmidt eigenvalues of reduced density matrices obtained from the ergodic eigenstates. We start by exploring the extreme value statistics of the ultrametric random matrices and then…

We compute the exact and limiting smallest eigenvalue distributions for two classes of $\beta$-Jacobi ensembles not covered by previous studies. In the general $\beta$ case, these distributions are given by multivariate hypergeometric…

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B…

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod…

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in…

We study the statistics of the largest eigenvalue lambda_max of N x N random matrices with unit variance, but power-law distributed entries, P(M_{ij})~ |M_{ij}|^{-1-mu}. When mu > 4, lambda_max converges to 2 with Tracy-Widom fluctuations…

Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit ${\cal…

We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and…

We present a large deviation principle at speed N for the largest eigenvalue of some additively deformed Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the…

For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R}$. We show that the density of this…

We characterize the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian $\beta$-ensemble of random matrix theory, including in particular the Gaussian orthogonal…

Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus,…

An exact analytical description of extreme intensity statistics in complex random states is derived. These states have the statistical properties of the Gaussian and Circular Unitary Ensemble eigenstates of random matrix theory. Although…

Large H-selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in…

For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting…

We consider an ensemble of large non-Hermitian random matrices of the form $\hat{H}+i\hat{A}_s$, where $\hat{H}$ and $\hat{A}_s$ are Hermitian statistically independent random $N\times N$ matrices. We demonstrate the existence of a new…

We give a short, operator-theoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n \times n Gaussian Unitary Ensemble in the large matrix limit n \to \infty. This is…