Estimating generalized Lyapunov exponents for products of random matrices

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 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.

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 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.

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.

The index of exponential growth is found for the products of random matrices with values on a solvable Lie algebra. The result is expressed via the eigenvalues of the terms.

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.

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.

We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max,+)-algebra (that is, the semiring (ℝ∪{−∞},max,+)), and also for semigroups of (max,+)-linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed.

The most straightforward way to design artificial structures is to use periodicity and leverage its potential mathematical and physical consequences to our favor. However, strict periodicity, which is usually employed in designing artificial materials, is not practically feasible and most often some degrees of randomness are introduced in the fabrication process, which degrade the performance. In this presentation, using the Furstenberg theory on the product of random matrices and Monte Carlo simulations we explore the effect of randomness in the transmission performance of periodic structures designed to realize density-near-zero (DNZ) metamaterials. We show that DNZ propagation is very sensitive to randomness, and we unravel the physics behind this sensitivity with analytical theory and full-wave simulations.

Stationarity, invariance, and ergodicity; discrete- and continuous-time ergodic theorems; moment and maximum inequalities; multivariate ergodic theorems; sample intensity of a random measure; subadditivity and products of random matrices; conditioning and ergodic decomposition; shift coupling and the invariant σ-field

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.

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

Random walks in a random environment are considered on the set of integers when the moving particle can go at most steps to the right and at most steps to the left in a unit of time. The transition probabilities for such a random walk from a point are determined by the vector . It is assumed that the sequence is a sequence of independent identically distributed random vectors. Asymptotic properties with probability 1 are investigated for such a random process. An invariance principle and the law of the iterated logarithm for a product of independent random matrices are proved as auxiliary results.

Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry–Esseen bound of optimal order n^{-1/2} for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and L^{p} ) distances. The results show that in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate n^{-1/2} cannot be reached, refuting a popular belief in the literature. This can be salvaged by simple methods. We reveal that in order to maintain the optimal speed of convergence n^{-1/2} , simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of GARCH models of any order, functionals of dynamical systems arising from stochastic differential equations, iterated random functions and many more.