Distribution of local Lyapunov exponents in spin-glass dynamics

We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the $d=1$ Heisenberg-Mattis spin glass (HMSG) at zero temperature, by means of a connection to a suitable version of the Fokker-Planck (F-P) equation. We consider the local Lyapunov exponents (LLE), in particular the case of {\em instantaneous} LLE. We establish a connection between the transfer-matrix recursion relation for the problem, and an F-P equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form (stationary) solutions to the F-P equation are in excellent accord with numerical simulations, for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for non-stationary conditions are derived from the F-P equation in a special limit (in which diffusive effects tend to vanish), and also shown to provide a close description to the corresponding numerical-simulation data.