# zbMATH — the first resource for mathematics

Lipschitz percolation. (English) Zbl 1193.60115
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
Full Text: