The random connection model in high dimensions
type: Item
instance of: scholarly article
title: The random connection model in high dimensions
zbMATH Open document ID: 0897.60096
DOI: 10.1016/S0167-7152(97)00008-4
author: Mathew D. Penrose
author: Anish Sarkar
author: Ronald Meester
published in: Statistics \& Probability Letters
publication date: 1998-10-11T00:00:00Z
review text: The random connection model is described as follows. Let \(X'= \{x_1,x_2, \dots\}\) be a homogeneous Poisson point process with intensity \(\lambda\) on \(\mathbb{R}^d\). The corresponding inhomogeneous process is \(X=\{x_0= 0\in\mathbb{R}^d\), \(x_1,x_2, \dots\}\). For \(g:(0, \infty] \to[0,1]\) a sufficiently quickly decreasing function, the points \(x_i\), \(x_j\in X\) are connected with probability \(g(| x_i- x_j|)\) independently on anything else. Now \(C\) is the cluster of the origin if \(C\) is the set of \(x_i\)'s that lie in the connected component of \(X\) that contains the origin. It is shown, roughly speaking, that, for \(d\) tending to infinity, the critical intensity \(\lambda_c^{(d)} (g)\) behaves like \(1/I_d (g)\) and the probability of the appearence of an infinite cluster of the origin behaves like \(1-\psi (\lambda I_d (g))\). Here \(\lambda_c^{(d)} (g)\) is the infimum of the intensities for which an infinite cluster of the origin appears with positive probability. Further, \(I_d(g)= \int_{\mathbb{R}^d} g(| x|) dx\) and \(\psi (\mu)\) is the extinction probability for a simple branching process with Poisson offspring distribution with mean \(\mu\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 1092052
zbMATH Keywords: random connection model
zbMATH Keywords: continuum percolation
zbMATH Keywords: critical intensity
zbMATH Keywords: high dimension asymptotics
zbMATH Keywords: extinction probability
reviewed by: Petr Holický
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1092052