The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1

Abstract

We introduce a new disorder regime for directed polymers with one space and one time dimension that is accessed by scaling the inverse temperature parameter βwith the length of the polymer n. We scale β_n := βn^{-α} for alpha non-negative. This scaling sits in between the usual weak disorder (β= 0) and strong disorder regimes (β> 0). The fluctuation exponents zeta for the polymer endpoint and χfor the free energy depend on αin this regime, with α= 0 corresponding to the usual polymer exponents ζ= 2/3, χ= 1/3 and α>= 1/4 corresponding to the simple random walk exponents ζ= 1/2, χ= 0. For 0 < α< 1/4 the exponents interpolate linearly between these two extremes. At α= 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.

4 pages, 1 figure. Added a more detailed description of scaling results in the critical regime. Improved simulation results.