The Lyapunov spectrum of a continuous product of random matrices

We present a functional integration method for the averaging of continuous products Pt of N x N random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum of Pt. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white-noise force field