Large deviations for the one-dimensional Edwards model

Abstract

In this article, we prove a large deviation principle for the empirical drift of a one-dimensional Brownian motion with self-repellence called the Edwards model. Our results extend earlier work in which a law of large numbers and a central limit theorem were derived. In the Edwards model, a path of length T receives a penalty e-βHT, where HT is the self-intersection local time of the path and β ∈ (0, ∞) is a parameter called the strength of self-repellence. We identify the rate function in the large deviation principle for the endpoint of the path as β 2/3I (β-1/3·), with I(·) given in terms of the principal eigenvalues of a one-parameter family of Sturm-Liouville operators. We show that there exist numbers 0 < b** < b* < ∞ such that (1) I is linearly decreasing on [0, b**], (2) I is real-analytic and strictly convex on (b**, ∞), (3) I is continuously differentiable at b** and (4) I has a unique zero at b**. (The latter fact identifies b* as the asymptotic drift of the endpoint.) The critical drift b** is associated with a crossover in the optimal strategy of the path: for b ≥ b** the path assumes local drift b during the full time T, while for 0 ≤ b