On the cycle expansion for the Lyapunov exponent of a product of random matrices

The cycle expansion of the thermodynamical zeta function for the Lyapunov exponent of a product of random matrices typically converges exponentially with the maximal cycle length ( Mainieri 1992 Phys. Rev. Lett. 68 1965). In this paper we show that the convergent exponents are given by the spectrum of a properly defined evolution operator, which describes how a steady distribution of vector direction is established under the action of random matrices. The exponential decay terms are automatically eliminated in the cycle expansion of the spectral determinant, which greatly accelerates the convergence provided all matrix elements are positive numbers. As a marginal case, the random Fibonacci series is studied in detail, and it is shown that this method is helpful.