Bernoulli products of matrices: A deterministic model

Abstract

The structure of the product of random matrices appearing as transfer matrices in several physical problems is studied. Making use of the fact that such matrices can be diagonalized by commuting matrices, we develop a graphical trick in order to control the properties of the product. The emergence of resonances gives a hint of the appearance of fluctuations and leads to a control of their amplitudes. We introduce a deterministic model (Tower model) reproducing the asymptotic random behaviour of the Bernoulli trials of such matrices. In particular the model is tested in computing Lyapunov exponents.