A generating function approach to the growth rate of random matrix products

TL;DR

This paper introduces a generating function approach to efficiently compute the growth rate of random matrix products, demonstrated through examples like the Ising model, offering a new analytical tool validated against numerical methods.

Contribution

It presents a novel generating function formalism and two analytic methods for calculating the growth rate of random matrix products, improving efficiency and ease of implementation.

Findings

Analytic methods match numerical results in large regimes.

The approach is demonstrated on the Ising model with random magnetic fields.

The formalism is efficient and easy to implement.

Abstract

Random matrix products arise in many science and engineering problems. An efficient evaluation of its growth rate is of great interest to researchers in diverse fields. In the current paper, we reformulate this problem with a generating function approach, based on which two analytic methods are proposed to compute the growth rate. The new formalism is demonstrated in a series of examples including an Ising model subject to on-site random magnetic fields, which seems very efficient and easy to implement. Through an extensive comparison with numerical computation, we see that the analytic results are valid in a regime of considerable size.