The continuum limit of some products of random matrices associated with renewing flows

TL;DR

This paper analyzes the continuum limit of products of random matrices associated with renewing flows, deriving explicit formulas for the generalized Lyapunov exponent using spectral analysis and elliptic integrals.

Contribution

It introduces a method to compute the generalized Lyapunov exponent for continuum limits of random matrix products, including explicit calculations for dimensions 2 and 3.

Findings

Explicit formula for growth rate in 2D using elliptic integrals

Spectral problem for modulus-dependent flow derived

First terms of Lyapunov exponent expansions computed for d=2,3

Abstract

We consider the continuum limit of some products of random matrices in $\text{SL}(d,{\mathbb R})$ that arise as discretisations of incompressible renewing flows -- that is, of flows corresponding to a divergence-free velocity field that takes independent, identically-distributed values in successive time intervals of duration proportional to $\tau$. The statistical properties of the product are encoded in its generalised Lyapunov exponent whose computation reduces to finding the leading eigenvalue of a certain transfer operator. In the continuum limit obtained by neglecting the terms of order $o(\tau^2)$, the transfer operator becomes a partial differential operator and, for a certain type of disorder which we call ``symmetric'', some calculations are feasible. For $d=2$, we compute the growth rate of the product in terms of complete elliptic integrals. By letting the elliptic modulus vary, we obtain a spectral problem, corresponding to a modulus-dependent random renewing flow, which may be viewed as a perturbation of the spectral problem for the angular Laplacian. In this way, we deduce expansions for the generalised Lyapunov exponent in ascending powers of the modulus. These expansions generalise to the case $d \ge 2$, and we compute the first few terms explicitly for $d \in \{2,3\}$.