2 Large Deviations for the One - Dimensional Edwards Model

In this paper we prove a large deviation principle for the empirical drift of a onedimensional Brownian motion with self-repellence called the Edwards model. Our results extend earlier work in which a law of large numbers, respectively, a central limit theorem were derived. In the Edwards model a path of length T receives a penalty eT , 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 3 I(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∗∗; (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 < b∗∗ it assumes local drift b∗∗ during time b ∗∗ +b 2b T and local drift −b∗∗ during the remaining time b∗∗−b 2b T . Thus, in the second regime the path makes an overshoot of size b ∗∗ −b 2 T in order to reduce its intersection local time. 2000 Mathematics Subject Classification. 60F05, 60F10, 60J55, 82D60.