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Nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices

Abstract:

We prove the moments of the directed polymer partition function GZ, using an exact position space renormalization group scheme on a hierarchical lattice. After sufficient iteration the characteristic function f(n)=ln〈GZn〉 of the probability ℘(Z) converges to a stable limit f*(n). For small n the limiting behavior is independent of the initial distribution, while for large n, f*(n) is completely determined by it and is thus nonuniversal. There is a smooth crossover between the two regimes for small effective dimensions, and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution. For large effective dimensions there is a sharp transition between the two regimes, and analytical continuation from integer moments to n→0 is not possible. Replica arguments can account for most features of the observed results. © 1993 Plenum Publishing Corporation.