Connectivity properties of Mandelbrot's percolation process
type: Item
instance of: scholarly article
title: Connectivity properties of Mandelbrot's percolation process
zbMATH Open document ID: 0621.60110
DOI: 10.1007/BF00319291
publication date: 1988-01-01T00:00:00Z
review text: In J. Fluid Mech. 62, 331-358 (1974; Zbl 0289.76031) \textit{B. Mandelbrot} introduced a process in \([0,1]^ 2\) which he called ''canonical curdling'' and later used in his book(s) on fractals to generate self-similar random sets with Hausdorff dimension \(D\in (0,2)\). In this paper we will study the connectivity or percolation properties of these sets, proving all of the claims he made in Sect. 23 of: 'The fractal geometry of nature' (1982; Zbl 0504.28001) and a new one that he did not anticipate: There is a probability \(p_ c\in (0,1)\) so that if \(p<p_ c\) then the set is ''dustlike'' i.e., the largest connected component is a point, whereas if \(p\geq p_ c\) (notice the \(=)\) opposing sides are connected with positive probability, and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles. More succinctly put the system has a first order phase transition.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4007422
zbMATH Keywords: self-similar random sets
zbMATH Keywords: percolation properties
zbMATH Keywords: unique unbounded connected component
zbMATH Keywords: first order phase transition
author: Jennifer T. Chayes
author: Lincoln Chayes
author: Rick Durrett
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4007422