Asymptotic geometry of high-density smooth-grained Boolean models in bounded domains

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X [t]) t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X [t;β], β≥0, defined as the Gibbsian modifications of X [t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X [t;β] is qualitatively very similar to that of X [t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X [t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.