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

Abstract

For the 2D matrix Langevin dynamics that correspond to the continuous-time limit of the products of some 2 × 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 thus be 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 empirical current of the Riccati process, and one then 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 one to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson localization models are described in order to explicitly obtain the first cumulants of the finite-size Lyapunov exponent.