The Annals of Probability

A stable marriage of Poisson and Lebesgue

Christopher Hoffman, Alexander E. Holroyd, and Yuval Peres

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Abstract

Let Ξ be a discrete set in ℝd. Call the elements of Ξ centers. The well-known Voronoi tessellation partitions ℝd into polyhedral regions (of varying sizes) by allocating each site of ℝd to the closest center. Here we study “fair” allocations of ℝd to Ξ in which the regions allocated to different centers have equal volumes.

We prove that if Ξ is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale–Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.)

We show that the region allocated to each center ξ is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than ξ. We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in d>1.

Article information

Source
Ann. Probab., Volume 34, Number 4 (2006), 1241-1272.

Dates
First available in Project Euclid: 19 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1158673318

Digital Object Identifier
doi:10.1214/009117906000000098

Mathematical Reviews number (MathSciNet)
MR2257646

Zentralblatt MATH identifier
1111.60008

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Stable marriage point process phase transition

Citation

Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval. A stable marriage of Poisson and Lebesgue. Ann. Probab. 34 (2006), no. 4, 1241--1272. doi:10.1214/009117906000000098. https://projecteuclid.org/euclid.aop/1158673318


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The Annals of Probability

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