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2022 The β-Delaunay tessellation II: the Gaussian limit tessellation
Anna Gusakova, Zakhar Kabluchko, Christoph Thäle
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Electron. J. Probab. 27: 1-33 (2022). DOI: 10.1214/22-EJP782


We study the weak convergence of β- and β-Delaunay tessellations in Rd1 that were introduced in part I of this paper, as β. The limiting stationary simplicial random tessellation, which is called the Gaussian-Delaunay tessellation, is characterized in terms of a space-time paraboloid hull process in Rd1×R. The latter object has previously appeared in the analysis of the number of shocks in the solution of the inviscid Burgers’ equation and the description of the local asymptotic geometry of Gaussian random polytopes. In this paper it is used to define a new stationary random simplicial tessellation in Rd1. As for the β- and β-Delaunay tessellation, the distribution of volume-power weighted typical cells in the Gaussian-Delaunay tessellation is explicitly identified, establishing thereby a new bridge to Gaussian random simplices. Also major geometric characteristics of these cells such as volume moments, expected angle sums and also the cell intensities of the Gaussian-Delaunay tessellation are investigated.

Funding Statement

ZK was supported by the DFG under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - Geometry - Structure. CT and ZK were supported by the DFG via the priority program Random Geometric Systems.


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Anna Gusakova. Zakhar Kabluchko. Christoph Thäle. "The β-Delaunay tessellation II: the Gaussian limit tessellation." Electron. J. Probab. 27 1 - 33, 2022.


Received: 3 March 2021; Accepted: 16 April 2022; Published: 2022
First available in Project Euclid: 10 May 2022

Digital Object Identifier: 10.1214/22-EJP782

Primary: 52A22 , 52B11 , 53C65 , 60D05 , 60F05 , 60F17 , 60G55

Keywords: angle sums , beta’-Delaunay tessellation , beta-Delaunay tessellation , Gaussian simplex , Gaussian-Delaunay tessellation , Laguerre tessellation , paraboloid convexity , paraboloid hull process , Poisson point process , Stochastic geometry , typical cell , weighted typical cell


Vol.27 • 2022
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