Lipschitz percolation

Abstract

We prove the existence of a (random) Lipschitz function $F:{\mathbb{Z}}^{d-1}\to {\mathbb{Z}}^{+}$ such that, for every $x\in {\mathbb{Z}}^{d-1}$, the site $(x,F(x))$ is open in a site percolation process on ${\mathbb{Z}}^d$. The Lipschitz constant may be taken to be 1 when the parameter $p$ of the percolation model is sufficiently close to 1.

Minor error corrected, and reference updated