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November 2001 Self-Attractive Random Plymers
Achim Klenke, Remco Van Der Hofstad
Ann. Appl. Probab. 11(4): 1079-1115 (November 2001). DOI: 10.1214/aoap/1015345396


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


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Achim Klenke. Remco Van Der Hofstad. "Self-Attractive Random Plymers." Ann. Appl. Probab. 11 (4) 1079 - 1115, November 2001.


Published: November 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1043.60016
MathSciNet: MR1878291
Digital Object Identifier: 10.1214/aoap/1015345396

Primary: 60F05 , 60J15 , 60J55.

Keywords: central limit theorem , Knight’s theorem for local times of simple random walk , Localization , phase transition , Repulsive and attractive interaction , spectral analysis

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.11 • No. 4 • November 2001
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