A replica approach to products of random matrices

TL;DR

This paper introduces a replica method to analyze products of random matrices, linking the free energy to replica symmetric representations and correlation functions to non-trivial representations, inspired by techniques from disordered Ising models.

Contribution

It adapts the replica trick and group representation techniques to study the spectral properties of products of random matrices, providing new insights into their free energy and correlations.

Findings

Replica symmetric representation yields the free energy or Lyapunov exponent.

Non-trivial representations correspond to specific correlation functions.

The approach connects random matrix products with disordered system techniques.

Abstract

We analyse products of random $R\times R$ matrices by means of a variant of the replica trick which was recently introduced for one-dimensional disordered Ising models. The replicated transfer matrix can be block-diagonalized with help of irreducible representations of the permutation group. We show that the free energy (or the Lyapunov exponent) of the product corresponds to the replica symmetric representation, whereas non-trivial representations correspond to certain correlation functions.