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

TL;DR

This paper introduces an intermediate disorder regime for directed polymers in 1+1 dimensions by scaling the inverse temperature with the polymer length, revealing a continuum of fluctuation behaviors between weak and strong disorder regimes.

Contribution

It defines a new scaling regime for directed polymers that interpolates between known disorder regimes and precisely characterizes the limiting distribution at a critical scaling exponent.

Findings

Exponents interpolate linearly between weak and strong disorder regimes.

Exact limiting distribution identified at the critical scaling exponent.

New disorder regime bridges the gap between classical regimes.

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 \beta with the length of the polymer n. We scale \beta_n := \beta n^{-\alpha} for alpha non-negative. This scaling sits in between the usual weak disorder (\beta = 0) and strong disorder regimes (\beta > 0). The fluctuation exponents zeta for the polymer endpoint and \chi for the free energy depend on \alpha in this regime, with \alpha = 0 corresponding to the usual polymer exponents \zeta = 2/3, \chi = 1/3 and \alpha >= 1/4 corresponding to the simple random walk exponents \zeta = 1/2, \chi = 0. For 0 < \alpha < 1/4 the exponents interpolate linearly between these two extremes. At \alpha = 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.