Supercritical self-avoiding walks are space-filling

Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel

Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 315-326.

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $\gamma$ between two points on the boundary of a finite subdomain of ${\mathbb{Z}}^d$ is proportional to ${\mu }^{-\text{length}\left(\gamma \right)}$. When $\mu$ is supercritical (i.e. $\mu <{\mu }_c$ where ${\mu }_c$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

Dans cet article, nous considérons le modèle suivant de marches auto-évitantes : la probabilité d’une trajectoire auto-évitante $\gamma$ entre deux points fixés d’un sous-domaine fini de ${\mathbb{Z}}^d$ est proportionnelle à ${\mu }^{-\text{length}\left(\gamma \right)}$. Lorsque le paramètre $\mu$ est supercritique (i.e. $\mu <{\mu }_c$ ou ${\mu }_c$ est la constante de connectivité du réseau), nous prouvons que la trajectoire aléatoire remplit l’espace lorsque l’on considère la limite d’échelle du modèle.

Classification: 60K35, 60C05, 82C41
Keywords: self avoiding walk, connective constant

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