Rigorous bounds and the replica method for products of random matrices

The authors first develop a method to obtain rigorous bounds for the Lyapounov exponent of products of a random matrix. When applied to a class of 1D problems. including localisation, it reproduces the correct scaling behaviour at the band edge and gives very good approximations of the prefactors. They then study analytically the successive moments of the distribution law for the trace of the random matrix product within the whole energy band. The band centre anomaly is found to affect the whole statistics of the problem and the exact anomalous value of the Lyapounov exponent is recovered through the replica trick.