Real Random Matrices Research Articles

Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

Finite free cumulants: Multiplicative convolutions, genus expansion and infinitesimal distributions

Given two polynomials p ( x ) , q ( x ) p(x), q(x) of degree d d , we give a combinatorial formula for the finite free cumulants of p ( x ) ⊠ d q ( x ) p(x)\boxtimes _d q(x) . We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that ⊠ d \boxtimes _d converges to ⊠ \boxtimes as d d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers.

Berry–Esseen bound and precise moderate deviations for products of random matrices

Let {(g_{n})_{n\geq 1}} be a sequence of independent and identically distributed (i.i.d.) {d\times d} real random matrices. For {n\geq 1} set {G_n = g_n \ldots g_1} . Given any starting point {x=\mathbb R v\in\mathbb{P}^{d-1}} , consider the Markov chain {X_n^x = \mathbb R G_n v } on the projective space {\mathbb P^{d-1}} and define the norm cocycle by {\sigma(G_n, x)= \log (|G_n v|/|v|)} , for an arbitrary norm {|\cdot|} on \smash{\mathbb R^{d}} . Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple {(X_n^x, \sigma(G_n, x))} . These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain {X_n^x} . Cramér-type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple {(X_n^x, \sigma(G_n, x))} with a target function {\varphi} on the Markov chain {X_n^x} .

A note on the universality of ESDs of inhomogeneous random matrices

In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and hard-to-verify Fourier domination assumption may be replaced simply by a natural uniform anti-concentration assumption. Along the way, we show that inhomogeneous complex random matrices, whose expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose entries (after symmetrization) are uniformly anti-concentrated at $0$ and infinity, typically have smallest singular value $\Omega(n^{-1/2})$. The rate $n^{-1/2}$ is sharp, and closes a gap in the literature. Our proofs closely follow recent works of Livshyts, and Livshyts, Tikhomirov, and Vershynin on inhomogeneous real random matrices. The new ingredient is a couple of anti-concentration inequalities for sums of independent, but not necessarily identically distributed, complex random variables, which may also be useful in other contexts.

Precise large deviation asymptotics for products of random matrices

Let (gn)n⩾1 be a sequence of independent identically distributed d×d real random matrices with Lyapunov exponent λ. For any starting point x on the unit sphere in Rd, we deal with the norm |Gnx|, where Gn≔gn…g1. The goal of this paper is to establish precise asymptotics for large deviation probabilities P(log|Gnx|⩾n(q+l)), where q>λ is fixed and l is vanishing as n→∞. We study both invertible matrices and positive matrices and give analogous results for the couple (Xnx,log|Gnx|) with target functions, where Xnx=Gnx∕|Gnx|. As applications we improve previous results on the large deviation principle for the matrix norm ‖Gn‖ and obtain a precise local limit theorem with large deviations.

On the Correlation Functions of the Characteristic Polynomials of Real Random Matrices with Independent Entries

On the Correlation Functions of the Characteristic Polynomials of Real Random Matrices with Independent Entries

Preserving topology while breaking chirality: from chiral orthogonal to anti-symmetric Hermitian ensemble

We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B|DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B|D). It enjoys the special feature that, depending on the matrix dimension , it has exactly zero-mode for even (odd), throughout the symmetry transition. This ‘topological protection’ is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrix-valued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skew-orthogonal polynomials, depending on the topological index . These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for , in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite .

NEW SPECTRAL STATISTICS FOR ENSEMBLES OF 2 × 2 REAL SYMMETRIC RANDOM MATRICES

We investigate spacing statistics for ensembles of various real random matrices where the matrix-elements have various Probability Distribution Function (PDF: <em>f(x)</em>) including Gaussian. For two modifications of 2 × 2 matrices with various PDFs, we derive the spacing distributions <em>p(s)</em> of adjacent energy eigenvalues. Nevertheless, they show the linear level repulsion near s = 0 as <em>αs</em> where <em>α</em> depends on the choice of the PDF. More interestingly when <em>f</em>(<em>x</em>) = <em>xe</em><sup>−x<sup>2</sup></sup> (<em>f</em>(0) = 0), we get cubic level repulsion near s = 0: <em>p(s)</em> ~ s<sup>3</sup>e<sup>−s<sup>2</sup></sup>.We also derive the distribution of eigenvalues <em>D</em>(ε) for these matrices.

Probability that product of real random matrices have all eigenvalues real tend to 1

In this note we consider products of real random matrices with fixed size with all entries as i.i.d. random variables. The product of such matrices has all eigenvalues real, with high probability. In other words, let X1,X2,… be i.i.d. k×k real matrices, whose entries are independent and identically distributed from probability measure μ. Let An=X1X2…Xn. Then it is conjectured that P(Anhasallrealeigenvalues)→1asn→∞. We show that the conjecture is true when μ has an atom.

Erratum to: The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We correct an error in Theorem 12 of the original article concerning the edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble.

Eigenvalue Attraction

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.

Real eigenvalues of non-Gaussian random matrices and their products

We study the properties of the eigenvalues of real random matrices and their products. It is known that when the matrix elements are Gaussian-distributed independent random variables, the fraction of real eigenvalues tends to unity as the number of matrices in the product increases. Here we present numerical evidence that this phenomenon is robust with respect to the probability distribution of matrix elements, and is therefore a general property that merits detailed investigation. Since the elements of the product matrix are no longer distributed as those of the single matrix nor they remain independent random variables, we study the role of these two factors in detail. We study numerically the properties of the Hadamard (or Schur) product of matrices and also the product of matrices whose entries are independent but have the same marginal distribution as that of normal products of matrices, and find that under repeated multiplication, the probability of all eigenvalues to be real increases in both cases, but saturates to a constant below unity showing that the correlations amongst the matrix elements are responsible for the approach to one. To investigate the role of the non-normal nature of the probability distributions, we present a thorough analytical treatment of the 2 × 2 single matrix for several standard distributions. Within the class of smooth distributions with zero mean and finite variance, our results indicate that the Gaussian distribution has the maximum probability of real eigenvalues, but the Cauchy distribution characterized by infinite variance is found to have a larger probability of real eigenvalues than the normal. We also find that for the two-dimensional single matrices, the probability of real eigenvalues lies in the range .

On a Generalization of the Elliptic Law for Random Matrices

We consider the products of $m\ge 2$ independent large real random matrices with independent vectors $(X_{jk}^{(q)},X_{kj}^{(q)})$ of entries. The entries $X_{jk}^{(q)},X_{kj}^{(q)}$ are correlated with $\rho=\mathbb E X_{jk}^{(q)}X_{kj}^{(q)}$. The limit distribution of the empirical spectral distribution of the eigenvalues of such products doesn't depend on $\rho$ and equals to the distribution of $m$th power of the random variable uniformly distributed on the unit disc.

The Ginibre evolution in the large-N limit

We analyse statistics of the real eigenvalues of gl(N, R)-valued Brownian motion (the Ginibre evolution) in the limit of large N. In particular, we calculate the limiting two-time correlation function of spin variables associated with real eigenvalues of the Ginibre evolution. We also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by Forrester and Nagao [“Eigenvalue statistics of the real Ginibre ensemble,” Phys. Rev. Lett. 99(5), 050603 (2007)] and Borodin and Sinclair [“The Ginibre ensemble of real random matrices and its scaling limits,” Commun. Math. Phys. 291(1), 177–224 (2009)].

On the number of real eigenvalues of products of random matrices and an application to quantum entanglement

The probability that there are k real eigenvalues for an n-dimensional real random matrix is known. Here, we study this for the case of products of independent random matrices. Relating the problem of the probability that the product of two real two-dimensional random matrices has real eigenvalues to an issue of optimal quantum entanglement, this is fully analytically solved. It is shown that in π/4 fraction of such products the eigenvalues are real. Being greater than the corresponding known probability () for a single matrix, it is shown numerically that the probability that all eigenvalues of a product of random matrices are real tends to unity as the number of matrices in the product increases indefinitely. Some other numerical explorations, including the expected number of real eigenvalues, are also presented, where an exponential approach of the expected number to the dimension of the matrix seems to hold.

Level spacings for weakly asymmetric real random matrices and application to two-color QCD with chemical potential

We consider antisymmetric perturbations of real symmetric matrices in the context of random matrix theory and two-color quantum chromodynamics. We investigate the level spacing distributions of eigenvalues that remain real or become complex conjugate pairs under the perturbation. We work out analytical surmises from small matrices and show that they describe the level spacings of large random matrices. As expected from symmetry arguments, these level spacings also apply to the overlap Dirac operator for two-color QCD with chemical potential.

A note on the Marchenko-Pastur law for a class of random matrices with dependent entries

We consider a class of real random matrices with dependent entries and show that the limiting empirical spectral distribution is given by the Marchenko-Pastur law. Additionally, we establish a rate of convergence of the expected empirical spectral distribution.

Truncations of random orthogonal matrices

Statistical properties of nonsymmetric real random matrices of size M, obtained as truncations of random orthogonal N×N matrices, are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong nonorthogonality, M/N=const, the behavior typical to real Ginibre ensemble is found. In the case M=N-L with fixed L, a universal distribution of resonance widths is recovered.

Averages over Ginibre's Ensemble of Random Real Matrices

AbstractWe give a method for computing the ensemble average of multiplicative class func-tions over the Gaussian ensemble of real asymmetric matrices. These averages areexpressed in terms of the Pfaffian of Gram-like antisymmetric matrices formed withrespect to a skew-symmetric inner product related to the class function. 1 Introduction In the 1960’s Ginibre introduced three statistical ensembles of matrices whose entries arechosen independently with Gaussian probability from (resp.) R, C, and Hamilton’s Quater-nions [4]. These ensembles are respectively labeled GinOE, GinUE and GinSE in analogywith their Hermitian counterparts. Ginibre introduced a physical analogy between GinUEand two-dimensional electrostatics, but gave no applications for the other two ensembles.Since their introduction, many applications have been found for GinOE and GinSE.Here we report a method for determining ensemble averages over GinOE of certainfunctions which are constant on similarity (conjugacy) classes. GinOE is the space of N×Nreal matrices R

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.