The constants in the CLT for the Edwards model

Abstract

The Edwards model in one dimension is a transformed path measure for one-dimensional Brownian motion discouraging selfintersections In van der Hofstad den Hollander and Konig preprint a central limit theorem CLT is proved for the uctuations of the endpoint of the path around its linear asymptotics In the present paper we study the constants appearing in this CLT which represent the mean and the variance and the exponential rate of the normalizing constant We prove that the variance is strictly smaller than which shows that the weak interaction limit is singular Furthermore we give a relation between the normalizing constant in the Edwards model and the normalizing constant in the weakly interacting DombJoyce model The DombJoyce model is the discrete analogue of the Edwards model based on simple random walk and is studied in van der Hofstad den Hollander and Konig preprint The proofs are based on bounds for the eigenvalues of a certain one-parameter family of SturmLiouville dierential operators These bounds are obtained by using the monotonicity of the zeroes of the eigenfunctions in combination with computer plots of the power series approximation of the eigenfunctions and exact error estimates of the power series approximation