Geometry of Lipschitz percolation
type: Item
instance of: scholarly article
title: Geometry of Lipschitz percolation
zbMATH Open document ID: 1255.60167
DOI: 10.1214/10-AIHP403
author: G. R. Grimmett
author: A. E. Holroyd
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2012-06-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.3762
full work available at URL: https://projecteuclid.org/euclid.aihp/1334148200
reviewed by: Nasir N. Ganikhodjaev
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 6042928
zbMATH Keywords: percolation
zbMATH Keywords: Lipschitz embedding
zbMATH Keywords: random surface
zbMATH Keywords: branching processes
zbMATH Keywords: total progeny
MaRDI profile type: Publication
arXiv ID: 1007.3762
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review text: 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}}
description: scientific article; zbMATH DE number 6042928