Self-Attractive Random Plymers

Abstract

We consider a repulsion–attraction model for a random polymer of finite lengthin $\mathbb{Z}^d$. Its law is that of a finite simple random walk path in $\mathbb{Z}^d$ receiving a penalty $3^{-2\beta}$ for every self-intersection, and a reward $e^{\gamma/d}$ for every pair of neighboring monomers. The nonnegative parameters $\beta$ and $\gamma$ measure the strength of repellence and attraction, respectively.

We show that for $\gamma > \beta$ the attraction dominates the repulsion; that is, with high probability the polymer is contained in a finite box whose size is independent of the length of the polymer. For $\gamma<\beta$ the behavior is different. We give a lower bound for the rate at which the polymer extends in space. Indeed, we show that the probability for the polymer consisting of $n$ monomers to be contained in a cube of side length $\varepsilon n^{1/d}$ tends to zero as $n$ tends to infinity.

In dimension $d = 1$ we can carry out a finer analysis. Our main result is that for $0 < \gamma \leq \beta -1/2\log 2$ the end-to-end distance of the polymer grows linearly and a central limit theorem holds.

It remains open to determine the behavior for $\gamma \in (\beta - 1/2\log 2, \beta]$ .