Narrow Results

Let x be a complex random variable with mean zero and bounded variance σ². Let N_n be a random matrix of order n with entries being i.i.d. copies of x. Let λ₁, …, λ_n be the eigenvalues of . Define the empirical spectral distributionμ_n of N_n by the formula

The following well-known conjecture has been open since the 1950's:

Circular Law Conjecture: μ_n converges to the uniform distribution μ_∞ over the unit disk as n tends to infinity.

We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices.

The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

The Dozier–Silverstein theorem asserts the almost sure convergence of the empirical spectral distribution of information plus noise matrices, i.e. perturbations of deterministic matrices whose spectral distribution converges (information matrices) by random matrices with i.i.d. entries (noise matrices). We show that a modification of the original proof given by Dozier and Silverstein allows to extend this result to more general noise matrices, in particular matrices with independent columns satisfying a natural concentration inequality for quadratic forms, matrices with independent entries, satisfying a Lindeberg-type condition (recovering a recent result by Xie), matrices with heavy-tailed entries in the domain of attraction of the Gaussian distribution and certain classes of matrices with dependencies among columns (generalizing those investigated recently by O'Rourke and containing matrices with exchangeable entries). As a corollary we obtain the circular law for random matrices with independent log-concave isotropic rows.

Let X be a random matrix whose pairs of entries X_jk and X_kj are correlated and vectors (X_jk, X_kj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that , for any j, k = 1, …, n and 𝔼 X_jkX_kj = ρ for 1 ≤ j < k ≤ n. Let M_n be a non-random n × n matrix with ‖M_n‖ ≤ Kn^Q, for some positive constants K > 0 and Q ≥ 0. Let s_n(X + M_n) denote the least singular value of the matrix X + M_n. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that

Let $G$ be an $N\times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij}\sim {𝒩}(0,1)$. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if ${\lambda }_1,\dots ,{\lambda }_{N_{ℝ}}$ are the real eigenvalues of $G$, then for any even polynomial function $P(x)$ and even $N=2n$, we have the convergence in distribution to a normal random variable

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by $n^{-1/2}$. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

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 nonzero variance ${\sigma }^2$. The strong circular law, proved by Tao and Vu, states that almost surely, as $n\to \infty$, the empirical spectral distribution of $N_n/(\sigma \sqrt{n})$ converges to the uniform distribution on the unit disc in $ℂ$.

A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix $xI-N_n/(\sigma \sqrt{n})$ (where $x\in ℂ$ is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix $M$ with operator norm at most $n^{3/4-𝜖}$ and for all $\eta \ge 0$,

Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the $\infty \to 2$ operator norm of heavy-tailed random matrices.

In this paper, we analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of random centrosymmetric matrices converge to a normal distribution. We find the exact expression of the variance of the limiting normal distribution via combinatorial arguments. Moreover, we argue that the limiting spectral distribution of properly scaled centrosymmetric matrices follows the circular law.

It was conjectured in the early 1950's that the empirical spectral distribution of an n × n matrix, of iid entries, normalized by a factor of , converges to the uniform distribution over the unit disc on the complex plane, which is called the circular law. Only a special case of the conjecture, where the entries of the matrix are standard complex Gaussian, is known. In this paper, this conjecture is proved under the existence of the sixth moment and some smoothness conditions. Some extensions and discussions are also presented.

In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the corresponding results for the complex case are also of interest, especially for researchers in Electrical and Electronic Engineering. Thus, we convert almost all results to the complex case, whenever possible. Only the latest results, including some new ones, are stated as theorems here. The main purpose of the paper is to show how important methodologies, or mathematical tools, have helped to develop the theory. Some unsolved problems are also stated.