2018/06/22 by Raphaël Cerf, Wei Zhou, Cerf, Raphaël +1 · 2 citations
Mathematics · #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1806.08576
Several passages have been enhanced, and the main result of our paper entitled "There is no isolated interface edge in very supercritical percolation" has been included
arxiv created 2019/06/21 · arxiv updated 2019/06/24
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 localised within a distance of order (ln |Λ|)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 Λ grows.