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July 2006 A stable marriage of Poisson and Lebesgue
Christopher Hoffman, Alexander E. Holroyd, Yuval Peres
Ann. Probab. 34(4): 1241-1272 (July 2006). DOI: 10.1214/009117906000000098


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.


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Christopher Hoffman. Alexander E. Holroyd. Yuval Peres. "A stable marriage of Poisson and Lebesgue." Ann. Probab. 34 (4) 1241 - 1272, July 2006.


Published: July 2006
First available in Project Euclid: 19 September 2006

zbMATH: 1111.60008
MathSciNet: MR2257646
Digital Object Identifier: 10.1214/009117906000000098

Primary: 60D05

Keywords: phase transition , point process , Stable marriage

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 4 • July 2006
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