Large deviations for the one-dimensional Edwards model

Details Large deviations for the one-dimensional Edwards model Large deviations for the one-dimensional Edwards model

2002-03-20

arxiv.org/abs/math/0203214v1

Mathematics

Probability

Scientific paper

In this paper 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, respectively, 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 $\beta^{\frac 23} I(\beta^{-\frac 13}\cdot)$, 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

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