SELF-AVOIDING WALKS AND MANIFOLDS IN RANDOM-ENVIRONMENTS

Self-avoiding walks (SAWs) and manifolds (SAMs) in random environments are studied using a combination of Lifshitz arguments and field-theoretic methods. The number of N-step SAWs starting at the origin, Z, is shown to be a broadly distributed quantity whose typical value, Ztyp, behaves as ZtypZexp(-cN below four dimensions. Here =2-d and Z is the average number of SAWs at the origin. On the other hand, the integer moments of Z are exponentially larger than the average, i.e., ZkZkexp[ck1/(k-1)N] for the range 1<k<kc. Similar results hold for SAMs. Within the field theory for SAWs the results for 1<k<kc arise from a fluctuation-driven first-order phase transition in the k-replicated theory. Above kc, Griffiths singularities control the moments of Z. © 1990 The American Physical Society.