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Abstract
Consider a 2dimensional 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 finiterange connection function. This paper is concerned with the asymptotic behaviour of the graph G(λ, s, φ) in the larges 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 

Article number  53 
Pages (fromto)  110 
Number of pages  10 
Journal  Electronic Communications in Probability 
Volume  27 
Early online date  26 Oct 2022 
DOIs  
Publication status  Published  31 Dec 2022 
Bibliographical note
Funding Information:*Supported by EPSRC grant EP/T028653/1.
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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Coverage and connectivity in stochastic geometry
Penrose, M. (PI)
Engineering and Physical Sciences Research Council
15/12/20 → 15/03/25
Project: Research council