The Lyapunov Spectrum of a Continuous Product of Random Matrices

TL;DR

This paper introduces a functional integration method to exactly compute the Lyapunov spectrum of continuous products of random matrices, with applications to disordered systems and passive scalar advection.

Contribution

It presents a novel analytical approach for calculating the Lyapunov spectrum of random matrix products, applicable to various disordered physical systems.

Findings

Exact computation of Lyapunov spectrum statistics

Method applicable to linear systems with white noise forces

Insights into passive scalar advection in random flows

Abstract

We expose a functional integration method for the averaging of continuous products $\hat{P}_t$ of $N\times N$ random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum of $\hat{P}_t$. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white noise force field.