Lyapunov exponent for products of Markovian random matrices

We analyze the effect of finite memory on the Lyapunov exponent of products of random matrices by considering Markov trials. We study three different cases of physical interest: the one-dimensional Anderson model with correlated random potentials, light propagation in media with correlated random optical indices, and a mimic of the deterministic chaos appearing in dynamical systems with few degrees of freedom. In general the Lyapunov exponent is found to have the same qualitative shape as the inverse of the correlation length of the Markov process. We, however, observe that this rough proportionality fails in some relevant situations. We explain this unexpected behavior in the localization problem by using simple arguments.