Related papers: Deformed single ring theorems

We obtain a tail bound for the least non-zero singular value of $A-z$ when $A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a given point $z_0$ in the bulk of the spectrum. The argument relies on a resolvent…

Let $N_n$ be an $n\times n$ complex random matrix, each of whose entries is an independent copy of a centered complex random variable $z$ with finite non-zero variance $\sigma^{2}$. The strong circular law, proved by Tao and Vu, states that…

There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1 > \dots > a_N$. The joint eigenvalue distribution of the $(N - 1)$ top-left principal submatrices of a random matrix…

Let $\Sigma_{g}$ be a closed surface of genus $g\geq 2$ and $\Gamma_{g}$ denote the fundamental group of $\Sigma_{g}$. We establish a generalization of Voiculescu's theorem on the asymptotic $*$-freeness of Haar unitary matrices from free…

We consider random symmetric matrices with independent entries distributed according to the Haar measure on $\mathbb{Z}_p$ for odd primes $p$ and derive the distribution of their canonical form with respect to several equivalence relations.…

We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral Random Matrix Theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in…

Let $m\in \mathbb{N}$, $\alpha\in[0,1]$, and $V$ be a 1-periodic complex-valued distribution in the negative Sobolev space $H^{-m\alpha}[0,1]$. The singular non-self-adjoint eigenvalue problem $D^{2m}u+V u=\lambda u$, $D=-i d/dx$, with…

Recall that a unitary in a tracial von Neumann algebra is Haar if $\tau(u^n)=0$ for all $n\in \mathbb{N}$. We introduce and study a new Borel equivalence relation $\sim_N$ on the set of Haar unitaries in a diffuse tracial von Neumann…

We consider the Brown measure of $a+\mathfrak{c}$, where $a$ lies in a commutative tracial von Neumann algebra $\mathcal{B}$ and $\mathfrak{c}$ is a $\mathcal{B}$-valued circular element. Under certain regularity conditions on $a$ and the…

The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $n\times n$ matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension $n$ grows to infinity. Consider an…

Let $\{T_{k}\}_{k=1}^{\infty}$ be a family of *--free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let…

For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $\nu_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $\nu_\mathcal{S}$ can be regarded as a…

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…

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of…

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in…

We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \infty)$, then…

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent…

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 develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such…

This work examines various statistical distributions in connection with random Vandermonde matrices and their extension to $d$--dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to…