Boolean models in hyperbolic space

2024/08/07 by Daniel Hug, Günter Last, Hug, Daniel +3 · 3 citations
Computer Science · Mathematics · #52A55 #60F05 #60G55 #Cellular Automata and Applications #FOS: Mathematics #Metric Geometry (math.MG) #Polynomial and algebraic computation #Primary 60D05 #Probability (math.PR) #Secondary 52A22 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.03890

openalex publication_date 2024/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The union of the particles of a stationary Poisson process of compact (convex) sets in Euclidean space is called Boolean model and is a classical topic of stochastic geometry. In this paper, Boolean models in hyperbolic space are considered, where one takes the union of the particles of a stationary Poisson process in the space of compact (convex) subsets of the hyperbolic space. Geometric functionals such as the volume of the intersection of the Boolean model with a compact convex observation window are studied. In particular, the asymptotic behavior for balls with increasing radii as observation windows is investigated. Exact and asymptotic formulas for expectations, variances, and covariances are shown and univariate and multivariate central limit theorems are derived. Compared to the Euclidean framework, some new phenomena can be observed.

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• Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells
• On strong sharp phase transition in the random connection model
• Wicksell's corpuscle problem in spaces of constant curvature

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