A replica approach to products of random matrices

We analyse products of random $R\times R$ matrices by means of a variant of the replica trick which was recently introduced for one-dimensional disordered Ising models. The replicated transfer matrix can be block-diagonalized with help of irreducible representations of the permutation group. We show that the free energy (or the Lyapunov exponent) of the product corresponds to the replica symmetric representation, whereas non-trivial representations correspond to certain correlation functions.