Singular behaviour of certain infinite products of random 2 × 2 matrices

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Abstract
The authors consider an infinite product of random matrices which appears in several physical problems, in particular the Ising chain in a random field. The random matrices depend analytically on a parameter epsilon in such a way that for = 0 they all commute. Under certain conditions they find that the Lyapunov index of this product behaves approximately as C 2α for to 0. Their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations. They give an expression for the exponent alpha in terms of the probability distribution of the matrices, and for the proportionality constant C in terms of the solutions of the reduced integral equations. In cases where exact results are available, agreement is obtained. The physical consequences for disordered one-dimensional systems are pointed out.

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