×

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 \({\lambda_c}(\phi )\).

MSC:

60C05 Combinatorial probability
60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Duminil-Copin, H., Sidoravicius, V. and Tassion, V. (2016). Absence of infinite cluster for critical Bernoulli percolation on slabs. Comm. Pure Appl. Math. 69, 1397-1411. · Zbl 1342.82076
[2] Heydenreich, M., van der Hofstad, R., Last, G. and Matzke, K. (2020). Lace expansion and mean-field behavior for the random connection model. 1908.11356.
[3] Last, G. and Penrose, M. (2018) Lectures on the Poisson process. Cambridge University Press. · Zbl 1392.60004
[4] Lichev, L., Lodewijks, B., Mitsche, D. and Schapira, B. (2022) Bernoulli percolation on the Random Geometric Graph. 2205.10923.
[5] Liggett, T.M., Schonmann, R.H. and Stacey, A.M. (1997). Domination by product measures. Ann. Probab. 25, 71-95. · Zbl 0882.60046
[6] Meester, R., Penrose, M. D. and Sarkar, A. (1997) The random connection model in high dimensions. Statist. Probab. Lett. 35, 145-153. · Zbl 0897.60096
[7] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press. · Zbl 0858.60092
[8] Penrose, M. (2003) Random Geometric Graphs. Oxford University Press, Oxford. · Zbl 1029.60007
[9] Penrose, M.D. (1996) Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab. 6, 528-544. · Zbl 0855.60096
[10] Penrose, M.D. (2016) Connectivity of soft random geometric graphs. Ann. Appl. Probab. 26, 986-1028. · Zbl 1339.05369
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.