Distribution of local Lyapunov exponents in spin-glass dynamics

We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the one-dimensional Heisenberg-Mattis spin glass (HMSG) at zero temperature by means of a connection to a suitable version of the Fokker-Planck (FP) equation. We consider the local Lyapunov exponents (LLEs), in particular, the case of instantaneous LLE. We establish a connection between the transfer-matrix recursion relation for the problem and an FP equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form (stationary) solutions to the FP equation are in excellent accord with numerical simulations for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for nonstationary conditions are derived from the FP 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.