A new look at the interfaces in percolation

Abstract

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 $\Lambda$, the interface between the top and the bottom of the box is typically localised within a distance of order $(\ln |\Lambda|)^2$ of the set of the pivotal edges. We prove also that, in our dynamical coupling, the typical speed of the pivotal edges remains bounded as the box $\Lambda$ grows.