Seminar 65. Infinite products of random elements of matrix groups

Infinite products of random elements of matrix groups arise naturally in probability theory, mathematical physics and other areas of mathematics. In many cases it is important to know the growth rate of such products. Furstenberg and Kesten proved that infinite products of random matrices have at most exponential growth rate, called the Lyapunov exponent, and afterwards Furstenberg proved that in most cases the Lyapunov exponent of an infinite product of random matrices is strictly bigger than zero. In a sense, one could say that theorems of Furstenberg and Kesten are analogous to classical limit theorems for random variables. We will talk about several applications of infinite products of random matrices, discuss proofs of Furstenberg's and Kesten's theorems, and mention further research directions, including very recent results.