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 $e^{-\beta H_T}$, where $ H_T$ is the self-intersection local time of the path and $\beta\in(0,\infty)$ 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 $\beta^{2/3} I(\beta^{-1/3}\cdot)$, with $I(\cdot)$ given in terms of the principal eigenvalues of a one-parameter family of Sturm--Liouville operators. We show that there exist numbers $0<b^{**}<b^*<\infty$ such that (1) I is linearly decreasing on $[0,b^{**}]$, (2) I is real-analytic and strictly convex on $(b^{**},\infty)$, (3) I is continuously differentiable at $b^{**}$ and\break (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 $b\geq b^{**}$ the path assumes local drift b during the full time T, while for $0\leq b<b^{**}$ it assumes local drift $b^{**}$ during time $\frac{b^{**}+b}{2b^{**}}T$ and local drift $-b^{**}$ during the remaining time $\frac{b^{**}-b}{2b^{**}}T$. Thus, in the second regime the path makes an overshoot of size $\frac{b^{**}-b}{2}T$ so as to reduce its intersection local time.