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, ${\mathit{Z}}_{\mathrm{typ}}$, behaves as ${\mathit{Z}}_{\mathrm{typ}}$∼〈Z〉exp(-${\mathit{cN}}^{\mathrm{\alpha }}$) 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., 〈${\mathit{Z}}^{\mathit{k}}$〉∼〈Z${\mathrm{〉}}^{\mathit{k}}$exp[${\mathit{ck}}^{1/\mathrm{\alpha }}$(k-1)N] for the range 1k${\mathit{k}}_{\mathit{c}}$. Similar results hold for SAM’s. Within the field theory for SAW’s the results for 1k${\mathit{k}}_{\mathit{c}}$ arise from a fluctuation-driven first-order phase transition in the k-replicated theory. Above ${\mathit{k}}_{\mathit{c}}$, Griffiths singularities control the moments of Z.