## 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 language | English |
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Article number | 53 |

Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Electronic Communications in Probability |

Volume | 27 |

Early online date | 26 Oct 2022 |

DOIs | |

Publication status | Published - 31 Dec 2022 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty