Self-avoiding walks and manifolds in random environments

Self-avoiding walks (SAW’s) and manifolds (SAM’s) in random environments are studied using a combination of Lifshitz arguments and field-theoretic methods. The number of N-step SAW’s starting at the origin, Z, is shown to be a broadly distributed quantity whose typical value, Z_typ, behaves as Z_typ∼〈Z〉exp(-cN^α) below four dimensions. Here α=2-dν and 〈Z〉 is the average number of SAW’s at the origin. On the other hand, the integer moments of Z are exponentially larger than the average, i.e., 〈Z^k〉∼〈Z〉^kexp[ck^1/α(k-1)N] for the range 1<k<k_c. Similar results hold for SAM’s. Within the field theory for SAW’s the results for 1<k<k_c arise from a fluctuation-driven first-order phase transition in the k-replicated theory. Above k_c, Griffiths singularities control the moments of Z.