Products of Complex Rectangular and Hermitian Random Matrices

In the study of random matrices, Lyapunov exponents characterize the rate of exponential growth of the singular values of product of increasing number of |$i.i.d$| random matrices of fixed order. These have been well studied objects in the literature. Only recently, absolute values of the eigenvalues of such products of random matrices have come under analytical study. The quantities that characterize the rate of exponential growth of the absolute values of eigenvalues have been named stability exponents, in order to distinguish them from that of the singular values. In the cases of Ginibre matrices and truncated Haar unitary matrices, the stability exponents have been observed to match with the Lyapunov exponents. In this article, we generalize this result to the case of all isotropic random matrices. We also derive the asymptotic joint probability distributions for the fluctuations of both the singular values and the absolute values of eigenvalues of product of increasing number of |$i.i.d$| real or complex isotropic random matrices of fixed order. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of this, we show the probability that the product of |$n$||$i.i.d$| real isotropic random matrices has all the eigenvalues real goes to one as |$n \to \infty$|⁠.

Many problems in physics can be reduced to product of random matrices [1,2]. Let us briefly discuss some cases. In the one dimensional disordered systems, i.e., with Hamiltonian containing random couplings and random fields (for example a disordered Ising chain) the free energy is related to the maximum Lyapunov exponent λ1 of the product of suitable random transfer matrices. Another example is given by the discretized Schrödinger equation on a one dimensional lattice with a random potential [2]. Indeed, one can write the equation in terms of product of random matrices and in this case λ1 is the inverse of the characteristic length of the localized wave functions.KeywordsLyapunov ExponentRandom MatriceAsymptotic DistributionRandom MatrixRandom PotentialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study asymptotic distributions of large dimensional random matrices of the\nform $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using\nfree probability and combinatorics of colored labeled noncrossing partitions.\nThese matrices are taken from the set of off-diagonal blocks of the family\n$\\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically\nfree, asymptotically free against the family of deterministic diagonal\nmatrices, and whose norms are uniformly bounded almost surely. This class\nincludes unitarily invariant Hermitian random matrices with limit distributions\ngiven by compactly supported probability measures $\\nu$ on the real line. We\nexpress the limit moments in terms of colored labeled noncrossing pair\npartitions, to which we assign weights depending on even free cumulants of\n$\\nu$ and on asymptotic dimensions of blocks (Gaussianization). For products of\n$p$ independent blocks, we show that the limit moments are linear combinations\nof a new family of polynomials called generalized multivariate Fuss-Narayana\npolynomials. In turn, the product of two blocks of the same matrix leads to an\nexample with rescaled Raney numbers.\n

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.

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by O(1∕n), which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.

We study the asymptotic behaviour of points under matrix cocyles generated by rectangular matrices. In particular we prove a random Perron-Frobenius and a Multiplicative Ergodic Theorem. We also provide an example where such products of random rectangular matrices arise in the theory of random walks in random environments and where the Multiplicative Ergodic Theorem can be used to investigate recurrence problems.

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far in all other parts of the spectrum the known standard universality classes have been identified. In the limit of infinite products the Lyapunov and stability exponents share the same normal distribution. To leading order they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.

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 > 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 > 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 > 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.

We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper, we use another version of the techniques to extend the results of [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] to the case where the entries of the data matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, justifies the mean field approximation in the infinite width limit for the deep untrained neural networks and the property of the macroscopic universality of random matrix theory in this case.

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ &gt; 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-symmetric matrices: Assume that for any n , the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε &gt; 0, and τ &gt; 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, where the symbol denotes the weak convergence of distribution when n → ∞, is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u , v ≤ 1. Then, with probability one, for almost all x , where λ k (Ξ n × n ) are eigenvalues, F ( x ) is a distribution function whose Stieltjes transform satisfies the relation G α ( x , y , t ), as a function of x , is a distribution function satisfying the regularized stochastic canonical equation K 3 [3, 23] at the points x of continuity, is a random real functional whose Laplace transform of one-dimensional distribution is equal to The integrand <jats:inli

We discuss several techniques for the evaluation of the generalized Lyapunov exponents which characterize the growth of products of random matrices in the large-deviation regime. A Monte Carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general-purpose numerical method which is both efficient and easy to implement. Alternative techniques complementing this method are presented. These include the computation of the generalized Lyapunov exponents by solving numerically an eigenvalue problem, and some asymptotic results corresponding to high-order moments of the matrix products. Taken together, the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized Lyapunov exponents in a broad range of applications. Their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows.

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived for s=3 and s=1/3. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions, is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

Hill’s equations arise in a wide variety of physical problems, and are specified by a natural frequency, a periodic forcing function, and a forcing strength parameter. This classic problem can be generalized by allowing the forcing strength q k , the frequency λ k , and the period (Δ τ) k of the forcing function to vary from cycle to cycle. The growth rates for the solutions are then given by the growth rates of a matrix transformation, under matrix multiplication, where the elements vary from cycle to cycle. Simplified models of such problems are given by products of 2 ×2 random matrices drawn from a given class.

On considere le produit Ψn de n matrices aleatoires M i.i.d. ayant les proprietes suivantes: la suite (Ψn)n est bornee et M possede un vecteur invariant deterministe (constant). On suppose que la probabilite pour que, sur le cercle unite, M possede une unique valeur propre simple en 1 soit non nulle. On montre que Ψn est la somme d’un processus fluctuant et d’un processus decroissant. Ce dernier tend vers zero presque surement et exponentiellement rapidement lorsque n tend vers l’infini. Le terme fluctuant converge en moyenne de Cesaro vers une limite caracterisee explicitement par le vecteur invariant deterministe et par les donnees spectrales associees a la valeur propre 1 de la matrice $\mathbb{E}[M]$. Aucune hypothese supplementaire n’est faite sur les matrices M; elles peuvent etre a valeurs complexes et pas necessairement inversibles. On applique les resultats generaux a deux classes de systemes dynamiques: les chaines de Markov inhomogenes avec matrices de transition aleatoires (matrices stochastiques), ainsi que les systemes quantiques avec interactions repetees et aleatoires. Dans les deux cas, on prouve des resultats d’ergodicite pour la dynamique et on determine les etats limites.

Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights. They appear in random matrix theory in the form of special determinantal point processes that are called multiple orthogonal polynomial (MOP) ensembles. The correlation kernel in such an ensemble is expressed in terms of the solution of a Riemann-Hilbert problem, that is of size (r + 1) × (r + 1) in the case of r weights. A number of models give rise to a MOP ensemble, and we discuss recent results on models of non-intersecting Brownian motions, Hermitian random matrices with external source, and the two matrix model. A novel feature in the asymptotic analysis of the latter two models is a vector equilibrium problem for two or more measures, that describes the limiting mean eigenvalue density. The vector equilibrium problems involve both an external field and an upper constraint.