Cycle expansion for the Lyapunov exponent of a product of random matrices

Abstract

Using cycle expansion for the thermodynamic zeta function, a formula is derived for the Lyapunov exponent of a product of random hyperbolic matrices chosen from a discrete set. This allows for an accurate numerical solution of the Ising model in one dimension with quenched disorder. The formula is compared with weak disorder expansions and with the microcanonical approximation and shown to apply to matrices with degenerate eigenvalues.