Random polymers on the complete graph

摘要

Consider directed polymers in a random environment on the complete graph of size N. This model can be formulated as a product of i.i.d. N x N random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path.