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November, 1993 A Variational Characterization of the Speed of a One-Dimensional Self- Repellent Random Walk
Andreas Greven, Frank den Hollander
Ann. Appl. Probab. 3(4): 1067-1099 (November, 1993). DOI: 10.1214/aoap/1177005273

Abstract

Let Qnα be the probability measure for an n-step random walk (0,S1,,Sn) on Z obtained by weighting simple random walk with a factor 1α for every self-intersection. This is a model for a one-dimensional polymer. We prove that for every α(0,1) there exists θ(α)(0,1) such that limnQnα(|Sn|n[θ(α)ε,θ(α)+ε])=1for everyε>0. We give a characterization of θ(α) in terms of the largest eigenvalue of a one-parameter family of N×N matrices. This allows us to prove that θ(α) is an analytic function of the strength α of the self-repellence. In addition to the speed we prove a limit law for the local times of the random walk. The techniques used enable us to treat more general forms of self-repellence involving multiple intersections. We formulate a partial differential inequality that is equivalent to αθ(α) being (strictly) increasing. The verification of this inequality remains open.

Citation

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Andreas Greven. Frank den Hollander. "A Variational Characterization of the Speed of a One-Dimensional Self- Repellent Random Walk." Ann. Appl. Probab. 3 (4) 1067 - 1099, November, 1993. https://doi.org/10.1214/aoap/1177005273

Information

Published: November, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0784.60094
MathSciNet: MR1241035
Digital Object Identifier: 10.1214/aoap/1177005273

Subjects:
Primary: 60K35
Secondary: 58E30 , 60F10 , 60J15

Keywords: large deviations , Polymer model , self-repellent random walk , variational problems

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.3 • No. 4 • November, 1993
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