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 functionf(n)=ln〈GZn〉 of the probability ℘(Z) converges to a stable limitf*(n). For smalln the limiting behavior is independent of the initial distribution, while for largen,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 ton→0 is not possible. Replica arguments can account for most features of the observed results.
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