Geometry of Lipschitz percolation

Abstract

We prove several facts concerning Lipschitz percolation, including the following. The critical probability pL for the existence of an open Lipschitz surface in site percolation on Zd with d ≥ 2 satisfies the improved bound pL ≤ 1-1/[8(d - 1)]. Whenever p > pL, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. For p sufficiently close to 1, the connected regions of Zd-1 above which the surface has height 2 or more exhibit stretched-exponential tail behaviour. The last statement is proved via a stochastic inequality stating that the lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Zd. © Association des Publications de l'Institut Henri Poincaré, 2012.