Geometry of Lipschitz percolation
The authors consider site percolation with parameter \(p\) on the lattice \(\mathbb{Z}^d\) with \(d\geq 2\) and prove the following facts concerning Lipschitz percolation. {\parindent=6mm \begin{itemize}\item[(1)] The critical probability \(p_L\) for the existence of an open Lipschitz surface satisfies the improved bound \(p_L\leq 1-1/[8(d-1)].\) \item[(2)] Whenever \(p>p_L\), the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. \item[(3)] For \(p\) sufficiently close to 1, the connected regions of \(\mathbb{Z}^{d-1}\) above which the surface has height \(2\) or more exhibit stretched-exponential tail behaviour.NEWLINENEWLINE\end{itemize}}
scientific article; zbMATH DE number 6042928
scientific article