Lipschitz percolation

Nicolas Dirr (University of Bath)
Patrick W. Dondl (University of Bonn)
Geoffrey R. Grimmett (Cambridge University)
Alexander E. Holroyd (Microsoft Research; University of British Columbia)
Michael Scheutzow (Technical University, Berlin)

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 $1$.

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Pages: 14-21

Publication Date: January 21, 2010

DOI: 10.1214/ECP.v15-1521

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