Related papers: The Elliptic Law

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…

In this paper we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral…

Let $\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix…

It is known that the empirical spectral distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this…

We study the spectral norm of large rectangular random Toeplitz and circulant matrices with independent entries. For Toeplitz matrices, we show that the scaled norm converges to the norm of a bilinear operator defined via the pointwise…

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $\tau \in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-\tau}$ and the…

In this paper we study the distribution of hitting times for a class of random dynamical systems. We prove that for invariant measures with super-polynomial decay of correlations hitting times to dynamically defined cylinders satisfy…

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…

We analyze the asymptotic behavior of random variables $x(n,x\_0)$ defined by $x(0,x\_0)=x\_0$ and $x(n+1,x\_0)=A(n)x(n,x\_0)$, where $\sAn$ is a stationary and ergodic sequence of random matrices with entries in the semi-ring…

We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in a sequence $X = (X_{i})$ of independent and identically distributed random variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a…

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a…

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian…

For two lacunary sequences $(M_{n,1})_{n\geq 2},(M_{n,2})_{n\geq 0}$ and suitable functions $f$ we introduce random matrix ensembles with \begin{equation*} X_{n,n'}=f(M_{n+n',1}x_1,M_{|n-n'|,2}x_2). \end{equation*} We prove weak convergence…

In this paper we study ensembles of random symmetric matrices $\X_n = {X_{ij}}_{i,j = 1}^n$ with dependent entries such that $\E X_{ij} = 0$, $\E X_{ij}^2 = \sigma_{ij}^2$, where $\sigma_{ij}$ may be different numbers. Assuming that the…

This papers contains two results concerning random $n \times n$ Bernoulli matrices. First, we show that with probability tending to one the determinant has absolute value $\sqrt {n!} \exp(O(\sqrt(n log n)))$. Next, we prove a new upper…

We revisit the regularity theory for uniformly elliptic equations.

We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of $n$ random points in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we establish the…

Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha$ for some fixed $0 < \alpha…

Let $\mathbf X$ be a random matrix whose pairs of entries $X_{jk}$ and $X_{kj}$ are correlated and vectors $ (X_{jk},X_{kj})$, for $1\le j<k\le n$, are mutually independent. Assume that the diagonal entries are independent from off-diagonal…

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow "band" around the unit circle of width proportional to $\frac1n$, where $n$ is the size of matrices. For general radially symmetric potentials…