Large deviations for the one-dimensional Edwards model

2002/03/20 by Remco van der Hofstad, R. van der Hofstad, van der Hofstad, R. +5
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F05 #60F10 #60J55 #82D60 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60F10 #msc:60J55 #msc:82D60

paper · pdf · doi:10.48550/arxiv.math/0203214

arxiv created 2002/03/20 · openalex publication_date 2002/03/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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^-βH_T, where H_T 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^{\frac 23} I(β^{-\frac 13}⋅), 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>

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