Self-Attractive Random Plymers

Van Der Hofstad, Remco ; Klenke, Achim

Ann. Appl. Probab., Tome 11 (2001) no. 2, p. 1079-1115 / Harvested from Project Euclid

Résumé

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 $\gamma \in (\beta - 1/2\log 2, \beta]$ .

Détail

Publié le : 2001-11-14
Classification:  Repulsive and attractive interaction,  phase transition,  Knight’s theorem for local times of simple random walk,  spectral analysis,  localization,  central limit theorem,  60F05,  60J15,  60J55.