Zeta function for the Lyapunov exponent of a product of random matrices.

A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is non-perturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and the heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formula is derived by using a Bernoulli dynamical system to mimic the randomness. The product of random matrices often appears in the study of disordered materials and of dynamical systems. The physical quantities of these systems are related to the rate of growth of the random product — the Lyapunov exponent. For example, in the study of an Ising model with quenched randomness the Lyapunov exponent is proportional to the free energy per particle; in the Schrödinger equation with a random potential, the Lyapunov exponent is proportional to the localization length of the wave function; and in the motion of a classical particle, the Lyapunov exponent indicates the degree of sensitivity to initial conditions (chaos). Since Dyson [1] studied a system of harmonic oscillators with random couplings, many problems have