An asymptotic result for Brownian polymers

TL;DR

This paper proves a conjecture about the long-term behavior of a Brownian polymer model with a specific repelling interaction, advancing understanding of its asymptotic shape.

Contribution

It establishes the asymptotic behavior of the continuous process representing the polymer's end for a certain class of repelling interactions, confirming a previous conjecture.

Findings

Proved the conjecture on asymptotic behavior of the process

Characterized the shape of the growing polymer over time

Extended previous models to include non-compact support interactions

Abstract

We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields 92 (1992) 337--349). We prove their conjecture about the asymptotic behavior of the underlying continuous process $X_t$ (corresponding to the location of the end of the polymer at time $t$) for a particular type of repelling interaction function without compact support.