A new look at the interfaces in percolation

Wei Zhou (1, 2) , Raphaël Cerf (3)

1 UP11 - Université Paris-Sud - Paris 11
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We propose a new definition of the interface in the context of the Bernoulli percolation model. We construct a coupling between two percolation configurations , one which is a standard percolation configuration, and one which is a percolation configuration conditioned on a disconnection event. We define the interface as the random set of the edges where these two configurations differ. We prove that, inside a cubic box Λ, the interface between the top and the bottom of the box is typically localized within a distance of order (ln |Λ|)^2 of the set of the pivotal edges.

Domaines: Probabilités [math.PR]

Wei Zhou, Raphaël Cerf. A new look at the interfaces in percolation. 2019. ⟨hal-02100893⟩