A lower bound on the box-counting dimension of crossings in fractal percolation
type: Item
instance of: scholarly article
title: A lower bound on the box-counting dimension of crossings in fractal percolation
zbMATH Open document ID: 0932.60085
DOI: 10.1016/S0304-4149(97)00117-8
author: M. E. Orzechowski
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: Following Mandelbrot, the fractal percolation process is introduced as an infinitely repeated subdivision of a square into \(M^2\) subsquares with the retention probability \(p\). The percolating path \(\Gamma \) joining the opposite edges of the original square exists with a non-zero probability if \(p\geq p_c\) and its lower \(\underline{\dim}_B(\Gamma)\) and upper \(\overline{\dim}_B(\Gamma)\) box dimensions are introduced in a usual way. The main result of the paper is the proof of a theorem giving an almost sure lower bound of \(\underline{\dim}_B(\Gamma)\) in the form \(1+v(M)z(z/|\log z|)^3, \;z=1-p\). The theorem is a considerable extension of a result obtained by \textit{L. Chayes} [ibid. 61, No.~1, 25-43 (1996; Zbl 0849.60092)], who proved the non-rectifiability of \(\Gamma \) and the existence of the lower bound of \(\underline{\dim}_B(\Gamma)\) in the form \(1+\zeta , \^^M\zeta >0,\) without specifying the dependence of \(\xi \) on \(M\) and \(p\) explicitly. Another theorem is stated giving an almost sure upper bound of \(\overline{\dim}_B(\Gamma)\) in the immediate vicinity of \(p=1.\) The detailed proof, given in the author's Ph. D. Thesis [University of St. Andrews, 1997], is only sketched here. Even when the both bounds are nontrivial, they are far from being tight.
reviewed by: Ivan Saxl
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 1364456
zbMATH Keywords: fractal percolation
zbMATH Keywords: Mandelbrot percolation
zbMATH Keywords: box-counting dimension
zbMATH Keywords: Hölder exponent
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1364456