Continuum percolation in the Gabriel graph

@article{Bertin2002ContinuumPI,
  title={Continuum percolation in the Gabriel graph},
  author={Etienne Bertin and Jean-Michel Billiot and R{\'e}my Drouilhet},
  journal={Advances in Applied Probability},
  year={2002},
  volume={34},
  pages={689 - 701}
}
In the present study, we establish the existence of site percolation in the Gabriel graph for Poisson and hard-core stationary point processes. 

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References

SHOWING 1-10 OF 22 REFERENCES

Existence of Delaunay Pairwise Gibbs Point Process with Superstable Component

The present stuffy deals with the existence of Delaunay pairwise Gibbs point process with superstable component by using the well-known Preston theorem. In particular, we prove the stability, the

Phase Transition and Percolation in Gibbsian Particle Models

We discuss the interrelation between phase transitions in interacting lattice or continuum models, and the existence of infinite clusters in suitable random-graph models. In particular, we describe a

Markov random fields and percolation on general graphs

Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation,

The analysis of the Widom-Rowlinson model by stochastic geometric methods

We study the continuum Widom-Rowlinson model of interpenetrating spheres. Using a new geometric representation for this system we provide a simple percolation-based proof of the phase transition. We

The random geometry of equilibrium phases

Phase transition in continuum Potts models

We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of

Packing Densities and Simulated Tempering for Hard Core Gibbs Point Processes

Simulated tempering is shown to be an efficient alternative to commonly used Markov chain Monte Carlo algorithms for hard core Gibbs point processes.

Spatial delaunay gibbs point processes

It is demonstrated that the Markov property is satisfied when interactions are only permitted for “Delaunay–cliques” and to further establish a Hammersley-Clifford type theorem for the Delaunay Gibbs point processes.

A proposal for the estimation of percolation thresholds in two-dimensional lattices

The authors propose a method to calculate percolation thresholds pc and their error bars Delta pc of two-dimensional (2D) lattices that enables them to estimate thresholds very accurately even when they use the MC data obtained from fairly small sizes.

Nearest-Neighbour Markov Point Processes and Random Sets

Summary The Markov point processes introduced by Ripley & Kelly are generalised by replacing fixed-range spatial interactions by interactions between neighbouring particles, where the neighbourhood