## 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

### Citations:

Zbl 0792.60097; Zbl 0810.60095
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### References:

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