Product of Random Matrices in a Microcanonical Ensemble

We give an approximate formula for the Lyapunov exponent of the product of N independent random matrices Mₖ all distributed according to the same probability distribution. We consider the case of a Bernoullian distribution, Mₖ=A with probability p and Mₖ=B otherwise. The method holds fixed the number of matrices of a given type, thus reducing fluctuations. We compare our analytic result to numerical calculations of the Lyapunov exponent for matrices A and B related to localization in the one-dimensional Anderson model where the approximation appears very precise for a wide range of energies. We give the fluctuation spectrum for the Lyapunov exponent also in terms of our approximation.