## Journal of Applied Probability

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

### Continuum AB percolation and AB random geometric graphs

#### 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 2*r*, 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