Random Transfer Matrices for the Overlap in Disordered Systems

A generating function is introduced to determine the probabilty P(q) of the overlap q in disordered systems via a product of random transfer matrices. In one-dimensional models, the overlap is obtained by the Lyapunov exponent λ of the product. Replica symmetry breaking at zero temperature corresponds to a discontinuity of the derivative of λ with respect to an appropriate coupling variable in the replica space. The method is illustrated in a frustrated magnetic model where q≠0