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Uniqueness of unbounded occupied and vacant components in Boolean models. (English) Zbl 0812.60093
There are considered Boolean models, generated by a stationary, ergodic point process in the \(d\)-dimensional Euclidean space \(R^ d\), where each point of the process is the center of a ball with random radius, which is independent of other balls and the point process. Thus \(R^ d\) is partitioned into two kinds of regions: occupied (by random balls) and vacant. The authors study the number of unbounded connected components in the occupied and vacant regions. They obtain sufficient conditions on the distribution of random radius for uniqueness of the occupied and vacant components. For a Boolean model, driven by a stationary Poisson point process, the uniqueness of the unbounded components is obtained without any condition at all. When the sufficient conditions are violated, examples of stationary, ergodic point processes, which admit more than one unbounded occupied or vacant components, are constructed. Finally the authors discuss, in brief, how to generalize the results to more general shapes than just balls.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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