Giant component of the soft random geometric graph

DOI10.1214/22-ECP491zbMATH Open1515.60048arXiv2204.10219MaRDI QIDQ6397100

Mathew D. Penrose

Publication date: 21 April 2022

Abstract: Consider a 2-dimensional soft random geometric graph $G(lambda,s,phi)$, obtained by placing a Poisson($lambda{s}^2$) number of vertices uniformly at random in a square of side $s$, with edges placed between each pair $x,y$ of vertices with probability $phi(|x-y|)$, where $$ is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph $G(lambda,s,phi)$ in the large-$s$ limit with $(lambda,phi)$ 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 $lambda$ equals the critical value $lambd{a}_c(phi)$.

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Interacting random processes; statistical mechanics type models; percolation theory (60K35) Combinatorial probability (60C05)

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