Self-Attractive Random Plymers

We consider a repulsion.attraction model for a random polymer of finite lengthin Zd. Its law is that of a finite simple random walk path in Zd receiving a penalty 3.2. for every self-intersection, and a reward e./d for every pair of neighboring monomers. The nonnegative parameters . and . measure the strength of repellence and attraction, respectively. We show that for .>. 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 .<. 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 .n1/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<....1/2log2 the end-to-end distance of the polymer grows linearly and a central limit theorem holds. It remains open to determine the behavior for ..(..1/2log2,.] .