Exact Lyapunov exponent for infinite products of random matrices

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 ...

Lyapunov Exponent for Products of Random Matrices

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...

Zeta function for the Lyapunov exponent of a product of random matrices.

A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is non-perturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and the heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formula is derived by using a Bernoulli dynamical sy...

Infinite Products of Random Matrices and Repeated Interaction Dynamics

Let Ψn be a product of n independent, identically distributed random matrices M , with the properties that Ψn is bounded in n, and that M has a deterministic (constant) invariant vector. Assuming that the probability of M having only the simple eigenvalue 1 on the unit circle does not vanish, we show that Ψn is the sum of a fluctuating and a decaying process. The latter converges to zero almost...