Circular Law For Random Matrices Research Articles

Fluctuations in the spectrum of non-Hermitian i.i.d. matrices

We consider large non-Hermitian random matrices X with independent identically distributed real or complex entries. In this paper, we review recent results about the eigenvalues of X: (i) universality of local eigenvalue statistics close to the edge of the spectrum of X [Cipolloni et al., “Edge universality for non-Hermitian random matrices,” Probab. Theory Relat. Fields 179, 1–28 (2021)], which is the non-Hermitian analog of the celebrated Tracy–Widom universality; (ii) central limit theorem for linear eigenvalue statistics of macroscopic test functions having 2 + ϵ derivatives [Cipolloni et al., “Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices,” Commun. Pure Appl. Math. (published online) (2021) and Cipolloni et al., “Fluctuation around the circular law for random matrices with real entries,” Electron. J. Probab. 26, 1–61 (2021)]. The main novel ingredients in the proof of these results are local laws for products of two resolvents of the Hermitization of X at two different spectral parameters, coupling of weakly dependent Dyson Brownian motions, and the lower tail estimate for the smallest singular value of X − z in the transitional regime |z| ≈ 1 [Cipolloni et al., “Optimal lower bound on the least singular value of the shifted Ginibre ensemble,” Probab. Math. Phys. 1, 101–146 (2020)].

Fluctuation around the circular law for random matrices with real entries

We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

Local laws for non-Hermitian random matrices and their products

We consider products of independent [Formula: see text] non-Hermitian random matrices [Formula: see text]. Assume that their entries, [Formula: see text], are independent identically distributed random variables with zero mean, unit variance. Götze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O’Rourke and Sochnikov [Products of independent non-Hermitian random matrices, Electron. J. Probab. 16 (2011) 2219–2245] proved that under these assumptions the empirical spectral distribution (ESD) of [Formula: see text] converges to the limiting distribution which coincides with the distribution of the [Formula: see text]th power of random variable uniformly distributed in the unit circle. In this paper, we provide a local version of this result. More precisely, assuming additionally that [Formula: see text] for some [Formula: see text], we prove that ESD of [Formula: see text] converges to the limiting distribution on the optimal scale up to [Formula: see text] (up to some logarithmic factor). Our results generalize the recent results of Bourgade et al. [Local circular law for random matrices, Probab. Theory Related Fields 159 (2014) 545–595], Tao and Vu [Random matrices: Universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015) 782–874] and Nemish [Local law for the product of independent non-hermitian random matrices with independent entries, Electron. J. Probab. 22 (2017) 1–35]. We also give further development of Stein’s type approach to estimate the Stieltjes transform of ESD.

Circular law for random matrices with exchangeable entries

AbstractAn 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 uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non‐Hermitian counterpart of a result of Chatterjee on the semi‐circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 454–479, 2016

Circular law for random matrices with unconditional log-concave distribution

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

Asymptotic distribution of complex zeros of random analytic functions

Let $\xi_0,\xi_1,\ldots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log(1+|\xi _0|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] where $f_{k,n}$ are deterministic complex coefficients. Let $\mu_n$ be the random measure counting the complex zeros of $\mathbf{G}_n$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}\mu_n$ converges in probability to some deterministic measure $\mu$ which is characterized in terms of the Legendre-Fenchel transform of $u$. The limiting measure $\mu$ does not depend on the distribution of the $\xi_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.

The local circular law III: general case

In the first part (Bourgade et al., Local circular law for random matrices, preprint, arXiv:1206.1449, 2012) of this article series, Bourgade, Yau and the author of this paper proved a local version of the circular law up to the finest scale $$N^{-1/2+ {\varepsilon }}$$ for non-Hermitian random matrices at any point $$z \in \mathbb {C}$$ with $$||z| - 1| > c $$ for any constant $$c>0$$ independent of the size of the matrix. In the second part (Bourgade et al., The local circular law II: the edge case, preprint, arXiv:1206.3187, 2012), they extended this result to include the edge case $$ |z|-1={{\mathrm{o}}}(1)$$ , under the main assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they proved that the circular law is valid near the spectral edge $$ |z|-1={{\mathrm{o}}}(1)$$ up to scale $$N^{-1/4+ {\varepsilon }}$$ .) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scale $$N^{-1/2+ {\varepsilon }}$$ for non-Hermitian random matrices at any point $$z \in \mathbb {C}$$ .

Local circular law for random matrices

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 unit circle. More precisely, if $$ | |z| - 1 | \ge \tau $$ for arbitrarily small $$\tau > 0$$ , the circular law is valid around $$z$$ up to scale $$N^{-1/2+ {\varepsilon }}$$ for any $${\varepsilon }> 0$$ under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

SOME REMARKS ON THE DOZIER–SILVERSTEIN THEOREM FOR RANDOM MATRICES WITH DEPENDENT ENTRIES

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.

The circular law for random matrices

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 unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.

The Circular Law. Twenty years later. Part III

In this paper we apply the REFORM method[3, 4] to the deduction of the system of canonical equations for normalized spectral functions of the matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is random matrix from class C 12 [26] or from the following class C 13 of distributions of random matrices: where are independent random real matrices whose entries are independent for every n, and for a certain δ &gt; 0 the Lyapunov condition is fulfilled: This problem has been considered in some publications for matrices A n + U n , where Unitary matrix U n from the class C 1 [26] is distributed by Haar measure, at the “ad hoc” level, without a strong proof, on the basis of heuristic calculations. Therefore, the behavior of limit n.s.f. of the sum of a random Unitary matrix U n and a nonrandom matrix A n has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13–27]) for random matrices A n + Ξ n B n , where Ξ n is the random matrix with independent entries, concerning, for example, the effectiveness of the REFORM method and the role of the martingale difference representation for the resolvent of random matrices. We give the formulation of the Circular Law for random matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is from the so called Class C 12 of random Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying spectral density p ( x, y ) of eigenvalues of A n + B n U n ( ε ) C n looks like several drops of mercury on a table, and inside of some drops it is possible that some dry circles appear. We call this support of limit density p ( x, y ) the Mercury support . If the distances between the centers of these drops are large enough we have several separate almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices [28, 29]. According to the distances between the centers of these drops, these drops might not touch each other or can merge creating fanciful shapes represented at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple Unitary matrix U n from class C 1 all its eigenvalues are distributed asymptotically uniformly on the circumference and, evidently, this limit distribution does not coincide with the uniform distribution on the circle. But if we will add to Unitary matrix U n any diagonal nonrandom matrix A n = ( δ ij a j ) then if the distances between diagonal entries a j are large enough we have almost the same picture of limit distribution of eigenvalues of U n + A n if Unitary matrix U n would be equal to the random matrix Ξ n with random independent entries satisfying the Global Circular Law (see Part I of this paper). The rough explanation of this phenomenon is the following: the expected scalar products of the vector row or vector column of matrix Ξ n have the order n −1 . Therefore, the matrix Ξ n looks like orthogonal random matrix. In this paper we use for matrices A n + B n U n ( ε ) C n the triply regularized V -transform[2, 5],( VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Randomly Independent Arrays ) where α &gt; 0, ε ≠= 0, γ ≠= 0 are regularization parameters , τ = t + is is a complex number and µ n { x,G n ( t, s, γ, ε )} is the normalized spectral function of G -matrix: Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for the limit distribution function of normalized spectral functions of some matrices A n + B n U n and the following V - domain G δ , δ &gt; 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of matrices A n + B n U n are found: where are eigenvalues of the matrix.