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

Products of random matrix products of $mathrm{SL}(2,mathbb{R})$, 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 $langle V^2rangle<infty$ and the case where its distribution presents a power law tail $p(V)sim|V|^{-1-alpha}$ for $0<alpha<2$. 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 $langle V^2rangle<infty$, one recovers Gaussian fluctuations with the variance equal to the mean value: $gamma_2simeqgamma_1$. For $langle V^2rangle=infty$, one finds $gamma_2simeq(2/alpha),gamma_1$ and non Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution $W(g)sim g^{-1+alpha/2}$ for $gto0$.