Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Supercritical self-avoiding walks are space-filling

Hugo Duminil-Copin, Gady Kozma, and Ariel Yadin

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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 $\mu^{-\mbox{length}(\gamma)}$. 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 $\mu^{-\mbox{length}(\gamma)}$. 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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 315-326.

First available in Project Euclid: 26 March 2014

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60C05: Combinatorial probability 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Self avoiding walk Connective constant


Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel. Supercritical self-avoiding walks are space-filling. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 315--326. doi:10.1214/12-AIHP528.

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