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I present a general framework allowing to carry out explicit calculation ofthe moment generating function of random matrix products$\Pi_n=M_nM_{n-1}\cdots M_1$, where $M_i$'s are i.i.d.. Following Tutubalin[Theor. Probab. Appl. {\bf 10}, 15 (1965)], the calculation of the generatingfunction is reduced to finding the largest eigenvalue of a certain transferoperator associated with a family of representations of the group. Theformalism is illustrated by considering products of random matrices from thegroup $\mathrm{SL}(2,\mathbb{R})$ where explicit calculations are possible. Forconcreteness, I study in detail transfer matrix products for theone-dimensional Schr\"odinger equation where the random potential is a L\'evynoise (derivative of a L\'evy process). In this case, I obtain a generalformula for the variance of $\ln||\Pi_n||$ and for the variance of$\ln|\psi(x)|$, where $\psi(x)$ is the wavefunction, in terms of a singleintegral involving the Fourier transform of the invariant density of the matrixproduct. Finally I discuss the continuum limit of random matrix products(matrices close to the identity ). In particular, I investigate a simple casewhere the spectral problem providing the generalized Lyapunov exponent can besolved exactly.

Fluctuations of the product of random matrices and generalized Lyapunov exponent