A Variational Characterization of the Speed of a One-Dimensional Self- Repellent Random Walk

Abstract

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 $\theta^\ast(\alpha) \in (0,1)$ such that $\lim_{n\rightarrow \infty} Q^\alpha_n\Bigg(\frac{|S_n|}{n} \in \lbrack\theta^\ast(\alpha) - \varepsilon,\theta^\ast(\alpha) + \varepsilon\rbrack\Bigg) = 1 \text{for every} \varepsilon > 0.$ We give a characterization of $\theta^\ast(\alpha)$ in terms of the largest eigenvalue of a one-parameter family of $\mathbb{N} \times \mathbb{N}$ matrices. This allows us to prove that $\theta^\ast(\alpha)$ 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 $\alpha \rightarrow \theta^\ast(\alpha)$ being (strictly) increasing. The verification of this inequality remains open.