Directed walks with complex random weights: phase diagram and replica symmetry breaking

(Received 26 February 1992, accepted in final form 21 April 1992)

Abstract
We consider the problem of directed walks (or polymers) in a random potential with both real and imaginary parts. The correlations of the potential are of short range. The problem is solved using a variational approximation which is expected to become exact in the limit of a large embedding dimension. A rich phase diagram is found which contains five phases. In addition to the simple high temperature phase, there are four ordered phases with different degrees of replica-symmetry-breaking. Three of the phases are dominated by interference (quantum) effects, and in two phases a pairing interaction between replicas plays an important role.