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

Products of random matrix products of $\mathrm{S}\mathrm{L}(2,\mathbb{R})$, 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 $⟨V^2⟩<\mathrm{\infty }$ and the case where its distribution presents a power law tail $p(V)\sim \mathrm{\mid }V{\mathrm{\mid }}^{-1-\alpha }$ 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 $⟨V^2⟩<\mathrm{\infty }$, one recovers Gaussian fluctuations with the variance equal to the mean value: ${\gamma }_2\simeq {\gamma }_1$. For $⟨V^2⟩=\mathrm{\infty }$, one finds ${\gamma }_2\simeq (2\mathrm{/}\alpha ){\gamma }_1$ and non Gaussian large deviations, related to the universal limiting behaviour of the conductance distribution $W(g)\sim g^{-1+\alpha/2}$for$g\to0$.