Lipschitz percolation

TL;DR

This paper proves the existence of a Lipschitz function mapping a lattice to positive integers that corresponds to open sites in a percolation process, with the Lipschitz constant approaching 1 as the percolation parameter nears 1.

Contribution

It establishes the existence of a Lipschitz surface in percolation models, extending understanding of geometric structures in high-dimensional percolation.

Findings

Existence of a Lipschitz function representing open sites

Lipschitz constant approaches 1 as p approaches 1

Provides a geometric characterization of percolation clusters

Abstract

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