Related papers: Circular Law for Random Block Band Matrices with G…
We consider inhomogeneous square random matrices of size $N$ with independent entries of mean 0 and finite variance. We assume that the variance profile of this matrix is doubly stochastic and has a band-like structure with an appropriately…
We consider the empirical eigenvalue distribution for a class of non-Hermitian random block tridiagonal matrices $T$ with independent entries. The matrix has $n$ blocks on the diagonal and each block has size $\ell_n$, so the whole matrix…
We consider the convergence of the ESD for non-Hermitian random band matrices with independent entries to the circular law, which is the uniform measure on the unit disk in the center of the complex plane. We assume that the bandwidth of…
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…
A well-known result in random matrix theory, proven by Kahn, Koml\'os and Szemer\'edi in 1995, states that a square random matrix with i.i.d. uniform $\{\pm 1\}$ entries is invertible with probability $1-\exp(-\Omega(n))$. As a natural…
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…
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…
In this paper, we derive limit laws for the empirical spectral distributions of random band and block matrices with correlated entries. In the first part of the paper, we study band matrices with approximately uncorrelated entries. We…
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…
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…
An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions…
Consider an $ N \times N$ Hermitian one-dimensional random band matrix with band width $W > N^{1 / 2 + \frak c} $ for any $ {\frak c} > 0$. In the bulk of the spectrum and in the large $ N $ limit, we obtain the following results: (i) The…
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…
For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random…
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…
Consider a complex random $n\times n$ matrix ${\bf X}_n=(x_{ij})_{n\times n}$, whose entries $x_{ij}$ are independent random variables with zero means and unit variances. It is well-known that Tao and Vu (Ann Probab 38: 2023-2065, 2010)…
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…
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…
It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing…
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…