Many-sided Poisson-Voronoi cells with only Gabriel neighbors

2024/03/12 by H. J. Hilhorst, Hilhorst, H. J.
Biochemistry, Genetics and Molecular Biology · Immunology and Microbiology · Materials Science · #Advanced biosensing and bioanalysis techniques #Dendrimers and Hyperbranched Polymers #FOS: Mathematics #FOS: Physical sciences #Immunotherapy and Immune Responses #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)

paper · pdf · doi:10.48550/arxiv.2403.07677

openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p_n^G be the probability for a planar Poisson-Voronoi cell to be n-sided \it and have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of log p_n^G in the limit of large n. We determine several statistical properties of a many-sided cell obeying this `Gabriel condition.' In particular, the cell perimeter, when parametrized as a function τ(θ) of the polar angle θ, behaves as a Brownian bridge on the interval 0≤θ≤ 2π. We point out similarities and differences with related problems in random geometry.

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