Large deviations for the one-dimensional Edwards model

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-(betaHT), where H-T is the self-intersection local time of the path and P E (0, 00) 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 beta(2/3)I(beta(-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* < infinity such that (1) 1 is linearly decreasing on [0, b**], (2) 1 is real-analytic and strictly convex on (b**, infinity), (3) I is continuously differentiable at b** and (4) 1 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 greater than or equal to b** the path assumes local drift b during the full time T, while for 0 < b < b** it assumes local drift b** during time b**+b/2bb** T and local drift -b** during the remaining time b**-b/2b** Thus, in the second regime the path makes an overshoot of size b**-b/2 T so as to reduce its intersection local time.