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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
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