Semianalytical solution of the random-product problem of matrices and discrete-time random evolution

Abstract

We consider the product of a large number of matrices chosen randomly (with some correlation) as $A$ or $B$. We introduce an exact functional equation, whose solution gives the limiting distribution and the Lyapunov exponent for the product of large number of matrices. Thus we suggest new semianalytical method to solve the random product problem of correlated matrices. The method can be easily generalized to high order matrices, as we deal with a larger systems of functional equations. We have provided a known numerical algorithm for the solution of the matrix product problem. We have also applied our method for a solution of discrete time random evolution. Our functional equation method provides exact transition point.