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 $1$.
Citation
Nicolas Dirr. Patrick Dondl. Geoffrey Grimmett. Alexander Holroyd. Michael Scheutzow. "Lipschitz percolation." Electron. Commun. Probab. 15 14 - 21, 2010. https://doi.org/10.1214/ECP.v15-1521
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