Conclusion and Suggestions for Future Works

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.

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 |$n \to \infty$|⁠.

In this thesis, we study infinite product of deterministic and random stochastic matrices. This mathematical object is one of the main analytical tools that is frequently used in various problems including: distributed computation, distributed optimization, distributed estimation, and distributed coordination. In many of these problems, a common goal is attempted to be achieved among a set of agents while there is no central coordination among them. A common theme for solving those problems is to reach a form of agreement by performing distributed averaging among the agents, which leads to an alternative way for the study of product of (deterministic) stochastic matrices, i.e. the study of weighted averaging dynamics.KeywordsRandom Stochastic MatricesNormal DynamicsMain Analytical ToolHegselmann-Krause ModelStochastic ChainThese 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 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 consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, , with the factors Mi drawn independently from an ensemble of random Markov matrices, whose columns are independent Dirichlet random variables. The statistical properties of the columns of , its largest eigenvalue and its spectrum are obtained exactly for N = 2 and numerically investigated for general N. For large t, the columns are Dirichlet-distributed, however the distribution is different from the initial one. As for the spectrum, we find that the eigenvalues converge to zero exponentially fast and investigate the statistics of the largest stability and Lyapunov exponents, which are approximated by Gamma distributions. We also observe a concentration of the spectrum on the real line for large t.

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.

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.

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 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.

Products and sums of random matrices have seen a rapid development in the\npast decade due to various analytical techniques available. Two of these are\nthe harmonic analysis approach and the concept of polynomial ensembles. Very\nrecently, it has been shown for products of real matrices with anti-symmetric\nmatrices of even dimension that the traditional harmonic analysis on matrix\ngroups developed by Harish-Chandra et al. needs to be modified when considering\nthe group action on general symmetric spaces of matrices. In the present work,\nwe consider the product of complex random matrices with Hermitian matrices, in\nparticular the former can be also rectangular while the latter has not to be\npositive definite and is considered as a fixed matrix as well as a random\nmatrix. This generalises an approach for products involving the Gaussian\nunitary ensemble (GUE) and circumvents the use there of non-compact group\nintegrals. We derive the joint probability density function of the real\neigenvalues and, additionally, prove transformation formulas for the\nbi-orthogonal functions and kernels.\n

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.

Exponential convergence of products of stochastic matrices

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.

Modelilng Unawareness in Arbitrary State Spaces