Electronic Communications in Probability

Poisson allocations with bounded connected cells

Alexander Holroyd and James Martin

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Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 69, 8 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320996

Digital Object Identifier
doi:10.1214/ECP.v20-3853

Mathematical Reviews number (MathSciNet)
MR3407213

Zentralblatt MATH identifier
1328.60026

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Holroyd, Alexander; Martin, James. Poisson allocations with bounded connected cells. Electron. Commun. Probab. 20 (2015), paper no. 69, 8 pp. doi:10.1214/ECP.v20-3853. https://projecteuclid.org/euclid.ecp/1465320996


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References

  • Chatterjee, Sourav; Peled, Ron; Peres, Yuval; Romik, Dan. Gravitational allocation to Poisson points. Ann. of Math. (2) 172 (2010), no. 1, 617–671.
  • Chatterjee, Sourav; Peled, Ron; Peres, Yuval; Romik, Dan. Phase transitions in gravitational allocation. Geom. Funct. Anal. 20 (2010), no. 4, 870–917.
  • Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval. A stable marriage of Poisson and Lebesgue. Ann. Probab. 34 (2006), no. 4, 1241–1272.
  • Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval. Tail bounds for the stable marriage of Poisson and Lebesgue. Canad. J. Math. 61 (2009), no. 6, 1279–1299.
  • Holroyd, Alexander E.; Pemantle, Robin; Peres, Yuval; Schramm, Oded. Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 1, 266–287.
  • Holroyd, Alexander E.; Peres, Yuval. Trees and matchings from point processes. Electron. Comm. Probab. 8 (2003), 17–27 (electronic).
  • Holroyd, Alexander E.; Peres, Yuval. Extra heads and invariant allocations. Ann. Probab. 33 (2005), no. 1, 31–52.
  • M. Huesmann. Optimal transport between random measures. 2012, arXiv:1206.3672. Ann. Inst. Henri Poincaré Probab. Stat., to appear.
  • Huesmann, Martin; Sturm, Karl-Theodor. Optimal transport from Lebesgue to Poisson. Ann. Probab. 41 (2013), no. 4, 2426–2478.
  • Krikun, Maxim. Connected allocation to Poisson points in $\Bbb R^ 2$. Electron. Comm. Probab. 12 (2007), 140–145.
  • R. Markó and A. Timár. A Poisson allocation of optimal tail. 2011. arXiv:1103.5259.
  • Nazarov, Fedor; Sodin, Mikhail; Volberg, Alexander. Transportation to random zeroes by the gradient flow. Geom. Funct. Anal. 17 (2007), no. 3, 887–935.
  • Soo, Terry. Translation-equivariant matchings of coin flips on $\Bbb Z^ d$. Adv. in Appl. Probab. 42 (2010), no. 1, 69–82.
  • A. Timar. Invariant matchings of exponential tail on coin flips in $\Bbb Z^ d$. 2009. arXiv:0909.1090.