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 Schr\"odinger equation with a random potential $V$, are studied. I consider both the case where the potential has a finite second moment $\langle V^2\rangle<\infty$ 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 $\langle V^2\rangle<\infty$, one recovers Gaussian fluctuations with the variance equal to the mean value: $\gamma_2\simeq\gamma_1$. For $\langle V^2\rangle=\infty$, one finds $\gamma_2\simeq(2/\alpha)\,\gamma_1$ and non Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution $W(g)\sim g^{-1+\alpha/2}$ for $g\to0$.