Related papers: Local Single Ring Theorem

Let $U$ and $V$ be two independent $N$ by $N$ random matrices that are distributed according to Haar measure on $U(N)$. Let $\Sigma$ be a non-negative deterministic $N$ by $N$ matrix. The single ring theorem [26] asserts that the empirical…

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical…

Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $\Sigma=\Sigma_N$ such that $A\to a$ and $\Sigma\to \sigma$ in $\ast$-distribution for some operators $a$ and $\sigma$ in a finite von…

We study the eigenvalues of non-normal square matrices of the form A_n=U_nT_nV_n with U_n,V_n independent Haar distributed on the unitary group and T_n real diagonal. We show that when the empirical measure of the eigenvalues of T_n…

We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical…

This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type $A+P$, where the rank of $P$ stays bounded as the dimension goes to infinity and where the matrix $A$ is a non Hermitian…

Recently, an analytic method was developed 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 existing Gaussian…

The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want…

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this…

We extend the so-called "single ring theorem"[1], also known as the Haagerup-Larsen theorem[2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the…

We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that…

We consider the least singular value of $M = R^* X T + U^* YV$, where $R,T,U,V$ are independent Haar-distributed unitary matrices and $X, Y$ are deterministic diagonal matrices. Under weak conditions on $X$ and $Y$, we show that the…

Given an orthogonally invariant probability measure on $GL(d,\mathbb{R})$, Mike Shub asked whether the average product of the $k$ top eigenvalues in the ensemble can be lower bounded by the average distortion along $k$ dimensional…

We prove a local law for eigenvalues of the random Hermitian matrices with external source $W_n=\frac{1}{n}X_n+A_n$ where $X_n$ is Wigner matrix and $A_n$ is diagonal matrix with only two values $a, -a$ on the diagonal. The local law is an…

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point $z$ away from the…

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices $A_{n}$ and $B_{n}$ rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix $U_{n}$ (i.e.…

Suppose $X$ is an $N \times n$ complex matrix whose entries are centered, independent, and identically distributed random variables with variance $1/n$ and whose fourth moment is of order ${\mathcal O}(n^{-2})$. In the first part of the…

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the…

We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with…

We consider an $n$ by $n$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix. We prove that for $k\sim n^{1/6}$ and $b^2:=\frac{1}{n}\operatorname{Tr}(|T|^2)$, as $n$…