Products Of Random Matrices Research Articles

Phase transitions for infinite products of large non-Hermitian random matrices

Phase transitions for infinite products of large non-Hermitian random matrices

Convergence of the Distributed SG Algorithm Under Cooperative Excitation Condition.

In this article, a distributed stochastic gradient (SG) algorithm is proposed where the estimators are aimed to collectively estimate an unknown time-invariant parameter from a set of noisy measurements obtained by distributed sensors. The proposed distributed SG algorithm combines the consensus strategy of the estimation of neighbors with the diffusion of regression vectors. For the theoretical investigation of the proposed algorithm, the main challenge lies in analyzing the influence of the Laplacian matrix on the state transition matrix and the properties of the product of nonindependent and nonstationary random matrices. Some analysis techniques such as graph theory and martingale theory are used to deal with the above issues. A cooperative excitation condition is introduced, under which the convergence of the distributed SG algorithm can be obtained without relying on the independency or stationarity assumptions of regression vectors which are commonly used in the existing literature. Furthermore, the convergence rate of the algorithm can be established. Finally, we show that all the sensors can cooperate to fulfill the estimation task even though any individual sensor cannot by a simulation example.

Fluctuations and correlations for products of real asymmetric random matrices

Fluctuations and correlations for products of real asymmetric random matrices

Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

Random dynamics on real and complex projective surfaces

Abstract We initiate the study of random iteration of automorphisms of real and complex projective surfaces, as well as compact Kähler surfaces, focusing on the classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.

Upper Bound on the Regularity of the Lyapunov Exponent for Random Products of Matrices

We prove that if {varvec{mu }} is a finitely supported measure on {varvec{SL}}_{textbf{2}}({mathbb {R}}) with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not {varvec{alpha }}-Hölder around {varvec{mu }} for any {varvec{alpha }} exceeding the Shannon entropy of {varvec{mu }} over the Lyapunov exponent of {varvec{mu }}.

Investigation of the Product of Random Matrices and Related Evolution Models

In this paper, we study the phase structure of the product of D * D order matrices. In each round, we randomly choose a matrix from a finite set of d matrices and multiply it with the product from the previous round. Initially, we derived a functional equation for the case of matrices with real eigenvalues and correlated choice of matrices, which led to the identification of several phases. Subsequently, we explored the case of uncorrelated choice of matrices and derived a simpler functional equation, again identifying multiple phases. In our investigation, we observed a phase with a smooth distribution in steady-state and phases with singularities. For the general case of D-dimensional matrices, we derived a formula for the phase transition point. Additionally, we solved a related evolution model. Moreover, we examined the relaxation dynamics of the considered models. In both the smooth phase and the phase with singularities, the relaxation is exponential. The superiority of relaxation in the smooth phase depends on the specific case.

On the generation of necklaces and bracelets in R

This note introduces a code snippet in R aiming to generate necklaces as well as bracelets. Among various uses, necklaces are useful tools to manage traces of products of random matrices. Functionality for necklaces and bracelets is provided with some examples of applications such as Lyndon words and de Bruijn sequences. The routines are collected in the Necklaces package available from the Comprehensive R Archive Network.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics.We start by obtaining new results about generating polynomials of lattice paths and total positivity of matrices and giving new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions with emphasis on its application to the analysis of multiple orthogonal polynomials. Then, we construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the branched continued fractions under study. Next, we introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we find an explicit recurrence relation satisfied by them. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials.Specialisations of the multiple orthogonal polynomials studied here include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to Nikishin systems of two measures involving modified Bessel functions, confluent hypergeometric functions, and Gauss' hypergeometric function, and multiple orthogonal polynomials with respect to Meijer G-functions used to investigate the singular values of products of Ginibre random matrices as well as a r-orthogonal polynomial sequence with constant recurrence coefficients for any positive integer r and particular instances of the Jacobi-Piñeiro polynomials.

Asymptotically linear iterated function systems on the real line

Given a sequence of i.i.d. random functions Ψn:R→R, n∈N, we consider the iterated function system and Markov chain, which is recursively defined by X0x:=x and Xnx:=Ψn−1(Xn−1x) for x∈R and n∈N. Under the two basic assumptions that the Ψn are a.s. continuous at any point in R and asymptotically linear at the “endpoints” ±∞, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie’s implicit renewal theory (Ann. Appl. Probab. (1991) 1 126–166) and can also be viewed as an adaptation of Kesten’s work on products of random matrices (Acta Math. (1973) 131 207–248) to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, for example, ARCH models and random logistic transforms.

Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ and denote by $$S_n:= g_n \cdots g_1$$ the associated random matrix product, where $$g_j$$ ’s are i.i.d.’s with law $$\mu $$ . We study statistical properties of random variables of the form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ where $$x \in \mathbb {P}^{d-1}$$ , $$\sigma $$ is the norm cocycle and u belongs to a class of admissible functions on $$\mathbb {P}^{d-1}$$ with values in $$\mathbb {R}\cup \{\pm \infty \}$$ . Assuming that $$\mu $$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $$\mathbb {R}$$ and Hölder continuous target functions on $$\mathbb {P}^{d-1}$$ . As particular cases, we obtain new limit theorems for $$\sigma (S_n,x)$$ and for the coefficients of $$S_n$$ .

Lyapunov Exponent, Universality and Phase Transition for Products of Random Matrices

Products of M i.i.d. random matrices of size \(N \times N\) are related to classical limit theorems in probability theory (\(N=1\) and large M), to Lyapunov exponents in dynamical systems (finite N and large M), and to universality in random matrix theory (finite M and large N). Under the two different limits of \(M\rightarrow \infty \) and \(N\rightarrow \infty \), the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both M and N go to infinity. This problem, proposed by Akemann et al. (J Phys A 47(39):395202, 2014) and Deift (SIGMA Symmetry Integr Geom Methods Appl 13, 2017), lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of M and N changes from 0 to \(\infty \): sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when \(M/N \rightarrow 0\), Gaussian fluctuation when \(M/N \rightarrow \infty \), and new critical phenomena when \(M/N \rightarrow \gamma \in (0,\infty )\). Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy–Widom distributions.

On *-Convergence of Schur–Hadamard Products of Independent Nonsymmetric Random Matrices

Abstract Let $\{x_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$ and $\{y_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$ be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix $X_n = ((x_{i - j}))_{1 \le i, j \le n}$ and a Hankel matrix $Y_n = ((y_{i + j}))_{1 \le i, j \le n}$, and let $M_n = X_n \odot Y_n$ be their elementwise/Schur–Hadamard product. In this article, we show that almost surely, $n^{-1/2}M_n$, as an element of the *-probability space $(\mathcal {M}_n({\mathbb {C}}), \frac {1}{n}\text {tr})$, converges in *-distribution to a circular variable. With i.i.d. Rademacher entries, this construction gives a matrix model for circular variables with only $O(n)$ bits of randomness. We also consider a dependent setup where $\{x_{\alpha }\}$ and $\{y_{\beta }\}$ are independent strongly multiplicative systems (à la Gaposhkin [7]) satisfying an additional admissibility condition, and have uniformly bounded moments of all orders—a nontrivial example of such a system being $\{\sqrt {2}\sin (2^n \pi U)\}_{n \in {\mathbb {Z}}_+}$, where $U \sim \textrm {Uniform}(0, 1)$. In this case, we show in-expectation and in-probability convergence of the *-moments of $n^{-1/2}M_n$ to those of a circular variable. Finally, we generalise our results to Schur–Hadamard products of structured random matrices of the form $X_n = ((x_{L_X(i, j)}))_{1 \le i, j \le n}$ and $Y_n = ((y_{L_Y(i, j)}))_{1 \le i, j \le n}$, under certain assumptions on the link-functions$L_X$ and $L_Y$, most notably the injectivity of the map $(i, j) \mapsto (L_X(i, j), L_Y(i, j))$. Based on numerical evidence, we conjecture that the circular law $\mu _{\textrm {circ}}$, that is, the uniform measure on the unit disk of ${\mathbb {C}}$, which is also the Brown measure of a circular variable, is in fact the limiting spectral measure (LSM)of $n^{-1/2}M_n$. If true, this would furnish an interesting example where a random matrix with only $O(n)$ bits of randomness has the circular law as its LSM.

Asymptotic freeness of unitary matrices in tensor product spaces for invariant states

In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In [6], the first and second authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalizations: (i) we state results of freeness in a context of general sequences of representations of the unitary group — the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.

Eigenvalue distribution of large random matrices arising in deep neural networks: Orthogonal case

This paper deals with the distribution of singular values of the input–output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.

Right large deviation principle for the top eigenvalue of the sum or product of invariant random matrices

In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices A and B as their dimensions goes to infinity. We consider a general framework containing the cases where A and/or B are taken from an invariant ensemble or are fixed diagonal matrices. We show that the tilting method introduced in Guionnet and Maïda (2020 Electron. J. Probab. 25 1–24) can be extended to our general setting and is equivalent to the study of a spherical spin glass model specific to the operation—sum of symmetric matrices/product of symmetric matrices/sum of rectangular matrices—we are considering.

A non-singular version of the Oseledeč ergodic theorem

AbstractKingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation T of a measure space $(X, \mu )$ . We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that $\mu $ and $\mu \circ T$ have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

Finite free convolutions of polynomials

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.

Hard edge statistics of products of Pólya ensembles and shifted GUE’s

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the Pólya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the Pólya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative Pólya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.