Density of states and Thouless formula for random unitary band matrices

Abstract.We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical sta-bility of certain quantum systems and can be considered as a unitary version of the Anderson model. It is also related with orthogonal polynomials on the unit circle. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.