Quermass-interaction Processes: Conditions for Stability

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 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. 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-deened: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle. The analysis of digital images and spatial patterns calls for tractable stochastic models of random sets and point processes. In this paper, we investigate new point process and germ-grain models which are constructed by weighting a Poisson point process (or germ-grain process) using exponentials of (sums of) quermass integrals (Minkowski functionals) of a Boolean model based on the reference random process. 2 These functionals are obtained from local geometric measurements including set volume and integrals of curvature over the boundary, and include the Euler-Poincar e characteristic. In the point process case the model under investigation generalises the Widom-Rowlinson penetrable spheres model 65] the area-interaction point process 4] and the morphological model in 37, 34, 38]. In this paper our main focus will be on the conditions under which planar quermass-interaction processes are stable in the sense of Ruelle (inequality (9) in Section 2.1 below). This is important because stability is an accessible condition for the density to be proper (to integrate to unit total mass rather than innnity), as well as being useful when studying the behaviour of the process (for example, whether its deeni-tion can be extended from bounded windows to the whole plane) and when devising simulation algorithms. Stability has already been established for the special case of area-interaction 4]; we shall establish it in greater generality, with particular attention to the Euler-Poincar e characteristic. Our arguments are basically geometric covering arguments of a rather non-standard form, essentially elementary but of some intrinsic geometric interest. In further papers we hope to develop inferential and simulation theory as well as to explore the utility …