A Variational characterisation of the speed of a one-dimensional self-repellent random walk - Greven, Andreas; den Hollander, Frank (Annals of Probability, 1993)
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 ope