Asymptotic Lyapunov Exponents for Large Random Matrices
نویسنده
چکیده
Suppose that A1, . . . , AN are independent random matrices whose atoms are iid copies of a random variable ξ of mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when ξ is gaussian then N−1 log ‖AN . . . A1‖ converges to a non-random limit. We extend this result to more general matrices with explicit rate of convergence. Our method relies on a simple connection between structures and dynamics.
منابع مشابه
Estimating generalized Lyapunov exponents for products of random matrices.
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 co...
متن کاملLoss of Memory of Random Functions of Markov Chains and Lyapunov Exponents
In this paper we prove that the asymptotic rate of exponential loss of memory of a random function of a Markov chain (Zt)t∈Z is bounded above by the difference of the first two Lyapunov exponents of a certain product of matrices. We also show that this bound is in fact realized, namely for almost all realization of the process (Zt)t∈Z, we can find symbols where the asymptotic exponential rate o...
متن کاملPositivity of Lyapunov exponents for Anderson-type models on two coupled strings
We study two models of Anderson-type random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed four-by-four symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol’dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in t...
متن کاملEstimating Lyapunov Exponents in Chaotic Time Series with Locally Weighted Regression
Nonlinear dynamical systems often exhibit chaos, which is characterized by sensitive dependence on initial values or more precisely by a positive Lyapunov exponent. Recognizing and quantifying chaos in time series represents an important step toward understanding the nature of random behavior and revealing the extent to which short-term forecasts may be improved. We will focus on the statistica...
متن کاملNumerical study of Lyapunov exponents for products of correlated random matrices.
We numerically study Lyapunov spectra and the maximal Lyapunov exponent (MLE) in products of real symplectic correlated random matrices, each of which is generated by a modified Bernoulli map. We can systematically investigate the influence of the correlation on the Lyapunov exponents because the statistical properties of the sequence generated by the map, whose correlation function shows power...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2016