Central limit theorem for a weakly interacting random polymer

Abstract

The Domb-Joyce model in one dimension is a transformed path measure for simple random walk on Zin which an n-step path gets a penalty e for every self-intersection. Here n is the strength of repellence, which may depend on n. We prove a central limit theorem for the end-to-end distance of the path in the case where n ! 0 and n 3 2 n ! 1 as n ! 1. It turns out that the mean grows like b 1 3 n n and the standard deviation like cpn, where b and c are constants that can be identied in terms of a Sturm-Liouville problem. The asymptotic mean shows an interpolation between ballistic behavior (n ) and diusive behavior (n = n ). Strikingly, the asymptotic standard deviation is independent of n. Our result is closely related to the central limit theorem for the Edwards model (the continuous space-time analogue of the Domb-Joyce model based on Brownian motion on R), which is proved in a separate paper.