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Register a new userFluctuations of the product of random matrices and generalized Lyapunov exponent
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Christophe Texier
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I present a general framework allowing to carry out explicit calculation of the 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 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 $mathrm{SL}(2,mathbb{R})$ 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.
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