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Quermass-interaction processes: conditions for stability

  • W. S. Kendall (a1), M. N. M. van Lieshout (a2) and A. J. Baddeley (a3)

Abstract

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

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Corresponding author

Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Email address: w.s.kendall@warwick.ac.uk
∗∗ Postal address: Centre for Mathematics and Computer Science, PO Box 94079, 1090 GB, Amsterdam, The Netherlands.
∗∗∗ Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia.

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Quermass-interaction processes: conditions for stability

  • W. S. Kendall (a1), M. N. M. van Lieshout (a2) and A. J. Baddeley (a3)

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