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.


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
Full Text: DOI


[1] Aldous, D. J. (1986). Self-intersections of 1-dimensional random walks. Probab. Theory Related Fields 72 559-587. · Zbl 0602.60055
[2] Bolthausen, E. (1990). On self-repellent one-dimensional random walks. Probab. Theory Related Fields 86 423-441. · Zbl 0691.60060
[3] Bry dges, D. C. and Spencer, T. (1985). Self-avoiding walk in 5 or more dimensions. Comm. Math. Phy s. 97 125-148. · Zbl 0575.60099
[4] Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities. Springer, Berlin. · Zbl 0146.38401
[5] Greven, A. and den Hollander, F. (1993). A variational characterization of the speed of a onedimensional self-repellent random walk. Ann. Appl. Probab. 3 1067-1099. · Zbl 0784.60094
[6] Knight, F. B. (1963). Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 109 56-86. JSTOR: · Zbl 0119.14604
[7] K önig, W. (1993). The drift of a one-dimensional self-avoiding random walk. Probab. Theory Related Fields 96 521-543. · Zbl 0792.60097
[8] K önig, W. (1994). The drift of a one-dimensional self-repellent random walk with bounded increments. Probab. Theory Related Fields 100 513-544. · Zbl 0810.60095
[9] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston. · Zbl 0780.60103
[10] Seneta, E. (1981). Non-Negative Matrices and Markov Chains. Springer, New York. · Zbl 0471.60001
[11] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton. · Zbl 0119.34304
[12] van der Hofstad, R. and den Hollander, F. (1995). Scaling for a random poly mer. Comm. Math. Phy s. 169 397-440. · Zbl 0821.60078
[13] van der Hofstad, R., den Hollander, F. and K önig, W. (1995). Central limit theorem for the Edwards model. Ann. Probab. · Zbl 0873.60009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.