A central limit theorem for a one-dimensional polymer measure

Abstract

Let $(S_n )_{n \in \mathbb {N}_0}$ be a random walk on the integers having bounded steps. The self-repellent (resp., self-avoiding) walk is a sequence of transformed path measures which discourage (resp., forbid) self-intersections. This is used as a model for polymers. Previously, we proved a law of large numbers; that is, we showed the convergence of $|S_n | /n$ toward a positive number $\theta$ under the polymer measure. The present paper proves a classical central limit theorem for the self-repellent and self-avoiding walks; that is, we prove the asymptotic normality of $(S_n - \theta_n) / \sqrt{n}$ for large n. The proof refines and continues results and techniques developed previously.