Lipschitz percolation. (English) Zbl 1193.60115

Dirr, Nicolas; Dondl, Patrick W.; Grimmett, Geoffrey R.; Holroyd, Alexander E.; Scheutzow, Michael

Electron. Commun. Probab. 15, 14-21 (2010).

Summary: 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.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Keywords:

percolation; Lipschitz embedding; random surface