Fluctuations of the product of random matrices and generalized Lyapunov exponent

I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products ${\mathrm{\Pi }}_n=M_n{M}_{n-1}\cdots M_1$, where $M_i$’s are i.i.d… Following Tutubalin [Theor. Probab. Appl. {\bf 10}, 15 (1965)], the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the group $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schr"odinger equation where the random potential is a L'evy noise (derivative of a L'evy process). In this case, I obtain a general formula for the variance of $\ln \mathrm{\mid }\mathrm{\mid }{\mathrm{\Pi }}_n\mathrm{\mid }\mathrm{\mid }$ and for the variance of $\ln \mathrm{\mid }\psi (x)\mathrm{\mid }$, where $\psi (x)$ is the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity ). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.