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2015 Poisson allocations with bounded connected cells
Alexander Holroyd, James Martin
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Electron. Commun. Probab. 20: 1-8 (2015). DOI: 10.1214/ECP.v20-3853


Given a homogenous Poisson point process in the plane, we prove that it is possible to partition the plane into bounded connected cells of equal volume, in a translation-invariant way, with each point of the process contained in exactly one cell. Moreover, the diameter $D$ of the cell containing the origin satisfies the essentially optimal tail bound $\mathbb{P}(D>r)<c/r$. We give two variants of the construction. The first has the curious property that any two cells are at positive distance from each other. In the second, any bounded region of the plane intersects only finitely many cells almost surely.


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Alexander Holroyd. James Martin. "Poisson allocations with bounded connected cells." Electron. Commun. Probab. 20 1 - 8, 2015.


Accepted: 26 September 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1328.60026
MathSciNet: MR3407213
Digital Object Identifier: 10.1214/ECP.v20-3853

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