## The Annals of Probability

### A Poisson allocation of optimal tail

#### Abstract

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).

#### Article information

Source
Ann. Probab. Volume 44, Number 2 (2016), 1285-1307.

Dates
Revised: December 2014
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.aop/1457960396

Digital Object Identifier
doi:10.1214/15-AOP1001

Mathematical Reviews number (MathSciNet)
MR3474472

Zentralblatt MATH identifier
1338.60027

#### Citation

Markó, Roland; Timár, Ádám. A Poisson allocation of optimal tail. Ann. Probab. 44 (2016), no. 2, 1285--1307. doi:10.1214/15-AOP1001. https://projecteuclid.org/euclid.aop/1457960396

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