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Supercritical self-avoiding walks are space-filling

Hugo Duminil-Copin; Gady Kozma; Ariel Yadin — 2014

Annales de l'I.H.P. Probabilités et statistiques

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $γ$ between two points on the boundary of a finite subdomain of $ℤ^{d}$ is proportional to $μ^{- length (γ)}$ . When $μ$ is supercritical (i.e. $μ l t; μ_{c}$ where $μ_{c}$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

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