Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models

TL;DR

This paper analyzes the large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics, providing two complementary methods and applying results to one-dimensional Anderson Localization models.

Contribution

It introduces two equivalent approaches for analyzing large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics and applies these to Anderson Localization.

Findings

Explicit expressions for the cumulant generating function of the Lyapunov exponent.

Demonstration of the equivalence between large deviations at level 2.5 and spectral analysis methods.

Explicit calculation of the first cumulants in Anderson Localization models.

Abstract

For the 2D matrix Langevin dynamics that corresponds to the continuous-time limit of the product of some $2 \times 2$ random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can be thus analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and of the empirical current of the Riccati process and one performs the appropriate Euler-Lagrange optimization in order to compute the cumulant generating function of the Lyapunov exponent. In the second approach, this cumulant generating function is obtained from the spectral analysis of the appropriate tilted Fokker-Planck operator. The associated conditioned process obtained via the generalization of Doob's h-transform allows to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson Localization models are described in order to obtain explicitly the first cumulants of the finite-size Lyapunov exponent.