Infinite Series of Singularities in the Correlated Random Matrices Product - Poghosyan, Ruben; Saakian, David B. (Frontiers in Physics, 2021)
We consider the product of a large number of two 2* 2 matrices chosen randomly (with some correlation): at any round there are transition probabilities for the matrix type, depending on the choice at previous round. Previously, a functional equation has been derived to calculate such a random product of matrices. Here, we identify the phase structure of the problem with exact expressions for the transition points separating ``localized'' and ``ergodic'' regimes. We demonstrate that the latter regime develops through a formation of an infinite series of singularities in the steady-state distribution of vectors that results from the action of the random product of matrices on an initial vector.