Giant component of the soft random geometric graph. (English) Zbl 1515.60048

Summary: 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 \(\phi :{\mathbb{R}_+}\to [0,1]\) 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 \({\lambda_c}(\phi )\).


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