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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