Abstract
The continuum percolation for Markov (or Gibbs) germ-grain models in dimension 2 is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincaré characteristic). We show that the percolation occurs for any coefficient of this linear combination and for a large enough activity parameter. An application to the phase transition of the multi-type quermass model is given.
Highlights
The germ-grain model is built by unifying random convex sets– the grains –centred at the points– the germs –of a spatial point process
Continuum percolation has been abundantly studied since the eighties and the pioneer paper of Hall [8]
When the grains are independent and identically distributed, and the germs are given by a Poisson point process (PPP), the germ-grain model is known as the Boolean model
Summary
The germ-grain model is built by unifying random convex sets– the grains –centred at the points– the germs –of a spatial point process. When the grains are independent and identically distributed, and the germs are given by a Poisson point process (PPP), the germ-grain model is known as the Boolean model In this context, continuum percolation is well-understood; see the book of Meester and Roy [13] for a very complete reference. The existence of a percolation threshold z∗ for the Boolean model relies on a basic (but essential) monotonicity argument: see [13], Chapter 2.2 This argument fails in the case of Gibbs point processes with Hamiltonian H. Following [2, 4], we use our percolation result (Theorem 1) to exhibit a phase transition phenomenon for Quermass interaction model with several type of particles (Theorem 2).
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