The shape of large balls in highly supercritical percolation

Anne-Laure Basdevant (1) , Nathanaël Enriquez (1, 2) , Lucas Gerin (1) , Jean-Baptiste Gouéré (3)

1 MODAL'X - Modélisation aléatoire de Paris X
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

We exploit a connection between distances in the infinite percolation cluster, when the parameter is close to one, and the discrete-time TASEP on $\mathbb{Z}$. This shows that when the parameter goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.

Mots clés: first-passage percolation; supercritical percolation; TASEP

Domaines: Probabilités [math.PR]

Anne-Laure Basdevant, Nathanaël Enriquez, Lucas Gerin, Jean-Baptiste Gouéré. The shape of large balls in highly supercritical percolation. Electronic Journal of Probability, 2014, pp.DOI: 10.1214/EJP.v19-3062. ⟨hal-00819221⟩