The Critical Attractive Random Polymer in Dimension One

Abstract

A polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e ^−β for every self-intersection and e ^γ/(2d) for every self-contact to the probability of an n-step simple random walk on ℤ^d, where β, γ>0 are parameters. It is known that for d=1 and γ>β the chain collapses down to finitely many sites, while for d=1 and γ<β it spreads out ballistically. Here we study for d=1 the critical case γ=β corresponding to the collapse transition and show that the end-to-end distance runs on the scale α _n =$\sqrt n$ (log n)^−1/4. We describe the asymptotic shape of the accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size α _n times a constant. Moreover, we derive the asymptotics of the partition function.