Supercritical self-avoiding walks are space-filling

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

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

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.

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.

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

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