The Annals of Probability

A stable marriage of Poisson and Lebesgue

Christopher Hoffman, Alexander E. Holroyd, and Yuval Peres

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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

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

First available in Project Euclid: 19 September 2006

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Zentralblatt MATH identifier

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

Stable marriage point process phase transition


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.

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  • Aldous, D. J. (2000). The percolation process on a tree where infinite clusters are frozen. Math. Proc. Cambridge Philos. Soc. 128 465--477.
  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29--66.
  • Carleson, L. (1967). Selected Problems on Exceptional Sets. Van Nostrand, Princeton, NJ.
  • Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Mathematical Association of America, Washington, DC.
  • Freire, M. V., Popov, S. and Vachkovskaia, M. (2005). Percolation for the stable marriage of Poisson and Lebesgue. Preprint. math.PR/0511186.
  • Gale, D. and Shapley, L. (1962). College admissions and stability of marriage. Amer. Math. Monthly 69 9--15.
  • Häggström, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423--1436.
  • Hoffman, C., Holroyd, A. E. and Peres, Y. (2005). Tail bounds for the stable marriage of Poisson and Lebesgue. Preprint. math.PR/0507324.
  • Holroyd, A. E. and Liggett, T. M. (2001). How to find an extra head: Optimal random shifts of Bernoulli and Poisson random fields. Ann. Probab. 29 1405--1425.
  • Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Probab. 33 31--52.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Knuth, D. E. (1997). Stable Marriage and Its Relation to Other Combinatorial Problems. Amer. Math. Soc., Providence, RI.
  • Liggett, T. M. (2002). Tagged particle distributions or how to choose a head at random. In In and Out of Equilibrium 133--162. Birkhäuser, Boston.
  • Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi diagrams, 2nd ed. Wiley, Chichester.
  • Lyons, R. and Peres, Y. (2006). Probability on Trees and Networks. Cambridge Univ. Press. Available at
  • Soardi, P. M. (1994). Potential Theory on Infinite Networks. Springer, Berlin.
  • Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057--2064.