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March 2016 A Poisson allocation of optimal tail
Roland Markó, Ádám Timár
Ann. Probab. 44(2): 1285-1307 (March 2016). DOI: 10.1214/15-AOP1001


The allocation problem for a $d$-dimensional Poisson point process is to find a way to partition the space to parts of equal size, and to assign the parts to the configuration points in a measurable, “deterministic” (equivariant) way. The goal is to make the diameter $R$ of the part assigned to a configuration point have fast decay. We present an algorithm for $d\geq3$ that achieves an $O(\operatorname{exp}(-cR^{d}))$ tail, which is optimal up to $c$. This improves the best previously known allocation rule, the gravitational allocation, which has an $\operatorname{exp}(-R^{1+o(1)})$ tail. The construction is based on the Ajtai–Komlós–Tusnády algorithm and uses the Gale–Shapley–Hoffman–Holroyd–Peres stable marriage scheme (as applied to allocation problems).


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Roland Markó. Ádám Timár. "A Poisson allocation of optimal tail." Ann. Probab. 44 (2) 1285 - 1307, March 2016.


Received: 1 March 2013; Revised: 1 December 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1338.60027
MathSciNet: MR3474472
Digital Object Identifier: 10.1214/15-AOP1001

Primary: 60D05

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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