On a multiscale continuous percolation model with unbounded defects
type: Item
instance of: scholarly article
title: On a multiscale continuous percolation model with unbounded defects
zbMATH Open document ID: 1056.60099
author: Mikhail V. Menshikov
author: S. Popov
author: M. Vachkovskaja
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2004-09-22T00:00:00Z
review text: The authors study multiscale percolation of unbounded Poisson Boolean models in dimension \(d \geq 2\). The Poisson Boolean model consists of the union of balls with centers following a Poisson process with intensity \(\lambda\) and independent radii following a distribution \(\rho\). The multiscale Poisson Boolean model consists of the union of independent and \(R\)-scaled Poisson Boolean models (level \(i\) having Poisson intensity \(\lambda R^{id}\) and radii distribution \(\rho R^{-i}\)). The main result of the paper is the extension of previously known results for bounded radii distributions to the case of an unbounded distribution \(\rho\). Under tail conditions on \(\rho\) the existence of a critical value \(\lambda_\rho\) is shown: for \(\lambda < \lambda_\rho\) there exists \(R_0\) such that for \(R \geq R_0\) there is no percolation. Another result of the paper is that if the expectation of \(\rho^{2\alpha d}\) is finite, then the expectation of the cluster to the power \(\alpha\) is also finite for small \(\lambda\).
reviewed by: Bo Markussen
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 2101598
zbMATH Keywords: multiscale percolation
zbMATH Keywords: Poisson Boolean model
zbMATH Keywords: unbounded radius
MaRDI profile type: Publication
full work available at URL: s00574-003-0022-3
OpenAlex ID: W1968560064
DOI: 10.1007/S00574-003-0022-3
description: scientific article; zbMATH DE number 2101598