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 Pi(n) = MnMn-1 ... M-1, where M-i's are i.i.d. Following Tutubalin (Theor Probab Appl 10(1):15-27, 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 SL(2, R) where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrodinger equation where the random potential is a Levy noise (derivative of a Levy process). In this case, I obtain a general formula 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 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.