16 Downloads (Pure)

Abstract

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c/(λ-λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.
Original languageEnglish
Pages (from-to)1228-1237
JournalJournal of Applied Probability
DOIs
Publication statusPublished - 1 Dec 2018

Cite this

On the critical threshold for continuum AB percolation. / Dereudre, David; Penrose, Mathew.

In: Journal of Applied Probability, 01.12.2018, p. 1228-1237.

Research output: Contribution to journalArticle

@article{7ace36c7bb3a4883b83526072d0ec946,
title = "On the critical threshold for continuum AB percolation",
abstract = "Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c/(λ-λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.",
author = "David Dereudre and Mathew Penrose",
year = "2018",
month = "12",
day = "1",
doi = "10.1017/jpr.2018.81",
language = "English",
pages = "1228--1237",
journal = "Journal of Applied Probability",
issn = "0021-9002",
publisher = "University of Sheffield",

}

TY - JOUR

T1 - On the critical threshold for continuum AB percolation

AU - Dereudre, David

AU - Penrose, Mathew

PY - 2018/12/1

Y1 - 2018/12/1

N2 - Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c/(λ-λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

AB - Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c/(λ-λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

UR - https://arxiv.org/abs/1712.04737

U2 - 10.1017/jpr.2018.81

DO - 10.1017/jpr.2018.81

M3 - Article

SP - 1228

EP - 1237

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

ER -