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Power diagrams and interaction processes for unions of discs. (English) Zbl 1146.60322
Summary: We study a flexible class of finite-disc process models with interaction between the discs. We let \(U\) denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of \(U\) such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only \(U\) is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of \(U\), and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of \(U\) and for simulating the disc process are discussed, and the software is made public available.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62M30 Inference from spatial processes
68U20 Simulation (MSC2010)
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