Estimating generalised Lyapunov exponents for products of random matrices

TL;DR

This paper introduces multiple techniques, including a Monte Carlo algorithm, for efficiently estimating generalized Lyapunov exponents of random matrix products, with applications demonstrated in fluid flow studies.

Contribution

It presents a new Monte Carlo importance sampling method and complementary approaches for computing generalized Lyapunov exponents, enhancing numerical evaluation capabilities.

Findings

Monte Carlo method is efficient and easy to implement.

Eigenvalue problem approach offers an alternative computation.

Asymptotic results aid in understanding high-order moments.

Abstract

We discuss several techniques for the evaluation of the generalised Lyapunov exponents which characterise 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 generalised 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 generalised 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.