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We investigate the level-density $\sigma(x)$ and level-spacing distribution $p(s)$ of random matrices $M=AF\neq M^{\dagger}$ where $F$ is a (diagonal) inner-product and $A$ is a random, real symmetric or complex Hermitian matrix with…

Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting $PT$-symmetry. Here we show that there is a one-to-one correspondence between complex $PT$-symmetric matrices and split-complex and…

This paper introduces a novel approach to probabilistic deep learning, kernel density matrices, which provide a simpler yet effective mechanism for representing joint probability distributions of both continuous and discrete random…

We review elementary properties of random matrices and discuss widely used mathematical methods for both hermitian and nonhermitian random matrix ensembles. Applications to a wide range of physics problems are summarized. This paper…

In applications of the density matrix renormalization group to nonhermitean problems, the choice of the density matrix is not uniquely prescribed by the algorithm. We demonstrate that for the recently introduced stochastic transfer matrix…

We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such…

We study the distribution of entries of a random permutation matrix under a "randomized basis," i.e., we conjugate the random permutation matrix by an independent random orthogonal matrix drawn from Haar measure. It is shown that under…

We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let $f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and $x…

We describe Generalized Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We will calculate a Laplace transform of such a density for finite…

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two…

In the course of studies of the measure of chaos for the distribution of the prime numbers among the positive integers N arched structures have been found. It is given a brief description of the fine structure of the positive integers…

We present randomized algorithms for estimating the trace and deter- minant of Hermitian positive semi-definite matrices. The algorithms are based on subspace iteration, and access the matrix only through matrix vector products. We analyse…

We apply the recently introduced method of hermitization to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the…

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic…

Consider an $n \times n$ non-Hermitian random matrix $M_n$ whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of $f(M_n)$ as $n$ tends to infinity, where…

The index of codivisibility of a set of integers is the size of its largest subset with a common prime divisor. For large random samples of integers, the index of codivisibility is approximately normal.

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these…

Let $A$ and $B$ be two $N$ by $N$ deterministic Hermitian matrices and let $U$ be an $N$ by $N$ Haar distributed unitary matrix. It is well known that the spectral distribution of the sum $H=A+UBU^*$ converges weakly to the free additive…

We consider dynamically defined Hermitian matrices generated from orbits of the doubling map. We prove that their spectra fall into the GUE universality class from random matrix theory.

We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified…