Lyapunov exponents: recent applications of Fürstenberg's theorem in spectral theory

In this talk we describe a new proof of a classical result and a number of new results that can be proved along similar lines. At the heart of this approach are the verification of the assumptions of F"urstenberg's theorem about products of random matrices and a way to parlay the output of this theorem via large deviation estimates into statements commonly referred to as ``Anderson localization.'' The various settings we consider include quantum evolution in random environments and orthogonal polynomials on the unit circle with random recursion parameters.