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A central limit theorem for a one-dimensional polymer measure. (English) Zbl 0862.60018

Summary: Let \((S_n)_{n\in\mathbb{N}_0}\) be a random walk on the integers having bounded steps. The self-repellent (resp., self-avoiding) walk is a sequence of transformed path measures which discourage (resp., forbid) self-intersections. This is used as a model for polymers. Previously, we proved a law of large numbers [Probab. Theory Relat. Fields 96, No. 4, 521-543 (1993; Zbl 0792.60097) and ibid. 100, No. 4, 513-544 (1994; Zbl 0810.60095)]; that is, we showed the convergence of \(|S_n|/n\) toward a positive number \(\Theta\) under the polymer measure. The present paper proves a classical central limit theorem for the self-repellent and self-avoiding walks; that is, we prove the asymptotic normality of \((S_n-\Theta n)\sqrt n\) for large \(n\). The proof refines and continues results and techniques developed previously.

MSC:

60F05 Central limit and other weak theorems
58E30 Variational principles in infinite-dimensional spaces
60F10 Large deviations
60G50 Sums of independent random variables; random walks
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References:

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