Related papers: Remarkable structures in integrable probability, I…
We study $n\times n$ random Hermitian matrix ensembles that are invariant under unitary conjugation. Let $I$ be a finite union of intervals lying in the bulk, and let $m_{k}^{(n)}$ be the $k$-th largest gap between consecutive eigenvalues…
The limiting distribution of eigenvalues of N x N random matrices has many applications. One of the most studied ensembles are real symmetric matrices with independent entries iidrv; the limiting rescaled spectral measure (LRSM)…
This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian…
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…
We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar…
We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}}…
We say a probability distribution $\mu$ is spectrally independent if an associated correlation matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if $\mu$ is spectrally…
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…
Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between L\'{e}vy measures to solve a…
Let $A$ be a real skew-symmetric Gaussian random matrix whose upper triangular elements are independently distributed according to the standard normal distribution. We provide the distribution of the largest singular value $\sigma_1$ of…
We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix…
Using high Reynolds number experimental data, we search for most dissipative, most intense structures. These structures possess a scaling predicted by log-Poisson model for the dissipation field $\epsilon_r$. The probability distribution…
The J\'{a}nossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The J\'{a}nossy density associated with an integrable kernel…
The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$…
Complex eigenvalues of random matrices $J=\text{GUE }+ i\gamma \diag (1, 0, \ldots, 0)$ provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the…
Conditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk…
In this paper, we study the largest eigenvalues of sample covariance matrices with elliptically distributed data. We consider the sample covariance matrix $Q=YY^*,$ where the data matrix $Y \in \mathbb{R}^{p \times n}$ contains i.i.d.…
We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such…
In the framework of Cramer's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(-exp(-x)) is the limit law for maximal gaps between…
We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We focus on Gaussian transformation families, which include matrix normal models and Gaussian graphical…