The Largest Liapunov Exponent for Random Matrices and Directed Polymers in a Random Environment

We study the largest Liapunov exponent for products of random matrices. The two classes of matrices considered are discrete, d-dimensional Laplacians, with random entries, and symplectic matrices that arise in the study of d-dimensional lattices of coupled, nonlinear oscillators. We derive bounds on this exponent for all dimensions, d, and we show that if d 3, and the randomness is not too strong, one can obtain an explicit formula for the largest exponent in the thermodynamic limit. Our method is based on an equivalence between this problem and the problem of directed polymers in a random environment. 2