A Variational characterisation of the speed of a one-dimensional self-repellent random walk
Greven A, den Hollander F (1993)
Publication Language: English
Publication Type: Journal article, Original article
Publication year: 1993
Journal
Publisher: Institute of Mathematical Statistics (IMS)
Book Volume: 3
Pages Range: 1067-1099
Journal Issue: 4
URI: https://projecteuclid.org/euclid.aoap/1177005273
Abstract
Let Qnα" role="presentation">Qαn be the probability measure for an n" role="presentation">n-step random walk (0,S1,…,Sn)" role="presentation">(0,S1,…,Sn) on Z" role="presentation">ℤ obtained by weighting simple random walk with a factor 1−α" role="presentation">1−α for every self-intersection. This is a model for a one-dimensional polymer. We prove that for every α∈(0,1)" role="presentation">α∈(0,1) there exists θ∗(α)∈(0,1)" role="presentation">θ∗(α)∈(0,1) such that limn→∞Qnα(|Sn|n∈[θ∗(α)−ε,θ∗(α)+ε])=1for everyε>0." role="presentation">limn→∞Qαn(|Sn|n∈[θ∗(α)−ε,θ∗(α)+ε])=1for everyε>0. We give a characterization of θ∗(α)" role="presentation">θ∗(α) in terms of the largest eigenvalue of a one-parameter family of N×N" role="presentation">ℕ×ℕ matrices. This allows us to prove that θ∗(α)" role="presentation">θ∗(α) is an analytic function of the strength α" role="presentation">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 α→θ∗(α)" role="presentation">α→θ∗(α) being (strictly) increasing. The verification of this inequality remains open.
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APA:
Greven, A., & den Hollander, F. (1993). A Variational characterisation of the speed of a one-dimensional self-repellent random walk. Annals of Probability, 3(4), 1067-1099. https://doi.org/10.1214/aoap/1177005273
MLA:
Greven, Andreas, and Frank den Hollander. "A Variational characterisation of the speed of a one-dimensional self-repellent random walk." Annals of Probability 3.4 (1993): 1067-1099.
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