Abstract

Consider a 2-dimensional soft random geometric graph G(λ, s, φ), obtained by placing a Poisson(λs 2) number of vertices uniformly at random in a square of side s, with edges placed between each pair x, y of vertices with probability φ(‖x − y‖), where φ: R + → [0, 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G(λ, s, φ) in the large-s limit with (λ, φ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λ c(φ).

Original languageEnglish
Article number53
Pages (from-to)1-10
Number of pages10
JournalElectronic Communications in Probability
Volume27
Early online date26 Oct 2022
DOIs
Publication statusPublished - 31 Dec 2022

Keywords

  • continuum percolation
  • random connection model
  • soft random geometric graph

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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