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

TL;DR

This work analyzes the continuum limit of products of random matrices in $SL(d,\mathbb{R})$ arising from renewing flows, focusing on the generalised Lyapunov exponent $L(\ell)$ and its computation via a transfer-operator framework. By exploiting a symmetric disorder assumption and the Kraichnan–Kazantsev limit, the transfer operator reduces to a differential operator whose spectrum yields cumulants and large-deviation statistics of trajectory separation; for $d=2$ the growth rate is expressed in terms of complete elliptic integrals and extended to a modulus-dependent spectral problem, with perturbative expansions in $k^2$ that generalize to $d=3$. The Iwasawa realisation and quasi-solvability facilitate explicit calculations, allowing closed-form, low-order expansions for the cumulants and explicit invariant subspaces in several cases. The results connect-renewing-flow matrix products to a continuum spectral problem, illuminating how disorder strength and pure-strain intervals shape Lyapunov statistics, with comparisons to prior work by Haynes and Vanneste and potential extensions beyond the continuum limit.

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 $τ$. 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 (Kraichnan--Kazantsev) limit obtained by neglecting the terms of order $o(τ^2)$, 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 $d \in \{2,3\}$.

Figures (2)

• Figure 1: Plots of $L$ (solid blue curve) and $L^\ast$ (solid red curve) against $\ell$ for $\tau=1$ and (a) $d=2$ and (b) $d=3$. In (a) are also displayed the approximation (\ref{['quadraticL']}) of $L$ (dashed curve) and the approximation (\ref{['quadraticLstar']}) of $L^\ast$ (dot-dashed curve) which use only the first two cumulants.
• Figure 2: Plots of the cumulants $\gamma_j$ against $1/d-k^2$ for $\tau=1$ and $1 \le j \le 4$. The blue and red curves correspond to $d=2$ and $d=3$ respectively: (a) $\gamma_1$ (solid curves) and $\gamma_2$ (dashed curves); (b) $\gamma_3$ (solid curves) and $\gamma_4$ (dashed curves).

Theorems & Definitions (10)

• Proposition 1.1
• Lemma 2.1
• Proposition 2.1
• Proposition 3.1
• Example 3.1
• Example 3.2
• Lemma 5.1
• Proposition 7.1
• Proposition A.1
• proof