Geometry of Lipschitz percolation
Probability
2010-07-23 v1
Abstract
We prove several facts concerning Lipschitz percolation, including the following. The critical probability p_L for the existence of an open Lipschitz surface in site percolation on Z^d with d\ge 2 satisfies the improved bound p_L \le 1-1/[8(d-1)]. Whenever p > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. The lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z^d. As a consequence, for p sufficiently close to 1, the connected regions of Z^{d-1} above which the surface has height 2 or more exhibit stretched-exponential tail behaviour.
Keywords
Cite
@article{arxiv.1007.3762,
title = {Geometry of Lipschitz percolation},
author = {Geoffrey R. Grimmett and Alexander E. Holroyd},
journal= {arXiv preprint arXiv:1007.3762},
year = {2010}
}