Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation

Products of random matrix products of , corresponding to transfer matrices for the one-dimensional Schrodinger equation with a random potential V, are studied. I consider both the case where the potential has a finite second moment and the case where its distribution presents a power law tail for . I study the generalized Lyapunov exponent of the random matrix product (i.e., the cumulant generating function of the logarithm of the wave function). In the high-energy/weak-disorder limit, it is shown to be given by a universal formula controlled by a unique scale (single parameter scaling). For , one recovers Gaussian fluctuations with the variance equal to the mean value: . For , one finds and non-Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution for .