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A variational characterization of the speed of a one-dimensional self- repellent random walk. (English) Zbl 0784.60094

Summary: Let \(Q^ \alpha_ n\) be the probability measure for an \(n\)-step random walk \((0,S_ 1,\ldots,S_ n)\) on \(\mathbb{Z}\) obtained by weighting simple random walk with a factor \(1-\alpha\) for every self-intersection. This is a model for a one-dimensional polymer. We prove that for every \(\alpha \in(0,1)\) there exists \(\theta^*(\alpha) \in(0,1)\) such that \[ \lim_{n \to \infty}Q^ \alpha_ n \left( | S_ n |/ n \in [\theta^*(\alpha)-\varepsilon,\;\theta^*(\alpha)+\varepsilon] \right) =1 \text{ for every } \varepsilon>0. \] We give a characterization of \(\theta^*(\alpha)\) in terms of the largest eigenvalue of a one- parameter family of \(\mathbb{N} \times \mathbb{N}\) matrices. This allows us to prove that \(\theta^*(\alpha)\) is an analytic function of the strength \(\alpha\) 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 \(\alpha \to \theta^*(\alpha)\) being (strictly) increasing. The verification of this inequality remains open.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
58E30 Variational principles in infinite-dimensional spaces
60F10 Large deviations
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