Journal of Applied Probability

Continuum AB percolation and AB random geometric graphs

Mathew D. Penrose

Abstract

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

Article information

Source
J. Appl. Probab., Volume 51A (2014), 333-344.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1417528484

Digital Object Identifier
doi:10.1239/jap/1417528484

Mathematical Reviews number (MathSciNet)
MR3317367

Zentralblatt MATH identifier
1315.60019

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C80: Random graphs [See also 60B20]
Secondary: 82B43: Percolation [See also 60K35] 05C40: Connectivity 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Bipartite geometric graph continuum percolation connectivity threshold

Citation

Penrose, Mathew D. Continuum AB percolation and AB random geometric graphs. J. Appl. Probab. 51A (2014), 333--344. doi:10.1239/jap/1417528484. https://projecteuclid.org/euclid.jap/1417528484


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