Estimating generalised Lyapunov exponents for products of random matrices

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.

arXiv commentRevised version: references added to the published version

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• Representation theory and products of random matrices in $\text{SL}(2,\mathbb{R})$
• Gaussian universality of $p$-adic random matrix products via corners
• Fast approximation of the $p$-radius, matrix pressure or generalised Lyapunov exponent for positive and dominated matrices
• A generating function approach to the growth rate of random matrix products