The distribution of Lyapunov exponents: Exact results for random matrices

Asymptotic Lyapunov Exponents for Large Random Matrices

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...

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...

Exact Lyapunov Exponent for Infinite Products of Random Matrices

In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2 × 2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the ...

Numerical 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...

Lyapunov Exponent for Products of Random Matrices