Related papers: Circular law for random matrices with exchangeable…

Let $\a$ be a complex random variable with mean zero and bounded variance $\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues of…

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the…

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le…

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…

Consider the empirical spectral distribution of complex random $n\times n$ matrix whose entries are independent and identically distributed random variables with mean zero and variance $1/n$. In this paper, via applying potential theory in…

We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n \times n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are…

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…

This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random…

The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$…

In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors $(X_{ij}, X_{ji})_{i \neq j}$ of entries. Under assumption of finite fourth moment of matrix entries it is proved that empirical…

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on…

Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n\times n$. We show that the empirical spectral distribution of the normalized matrix $\sqrt{n}(X-{\mathbf {E}}X)$ converges almost surely to the…

The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the…

The circular law asserts that if $X_n$ is a $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges almost surely to the uniform…

We consider non-Hermitian random matrices $X \in \mathbb{C}^{n \times n}$ with general decaying correlations between their entries. For large $n$, the empirical spectral distribution is well approximated by a deterministic density,…

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…

We show that, under some general assumptions on the entries of a random complex $n \times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges to the uniform law of an ellipsoid as $n$ tends to…

We consider the problem of determining the limiting spectral distribution for random matrices whose row distributions are permitted to have limited dependence. We assume mild moment conditions and give an extension of the…

For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, ..., X_m$ with i.i.d. centered entries with a finite $(2+\eta)$-th moment, $ \eta>0.$ As $n$ tends to infinity, we show that the empirical…

Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\la_{n,1},...,\la_{n,n}$ be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law…