2018/11/29 by Raphaël Cerf, Wei Zhou, Cerf, Raphaël +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1811.12368
There was a mistake in a construction of the first version in the section 3. So we withdraw the paper, and we have included the main result of this first version in the second version of our paper entitled "A new look at the interfaces in percolation" arXiv:1806.08576
arxiv created 2019/07/03 · arxiv updated 2019/07/04
We consider the Bernoulli bond percolation model in a box Λ (not necessarily parallel to the directions of the lattice) in the regime where the percolation parameter is close to 1. We condition the configuration on the event that two opposite faces of the box are disconnected. We couple this configuration with an unconstrained percolation configuration. The interface edges are the edges which differ in the two configurations. We prove that, typically, each interface edge is within a distance of order ln|Λ| of another interface edge or of a pivotal edge. We derive an estimate for the law of an edge which is far from the cut and the interface edges.