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A Poisson allocation of optimal tail

By Roland Markó and Ádám Timár

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-Koml\'{o}s-Tusn\'{a}dy algorithm and uses the Gale-Shapley-Hoffman-Holroyd-Peres stable marriage scheme (as applied to allocation problems).Comment: Published at http://dx.doi.org/10.1214/15-AOP1001 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Topics: Mathematics - Probability
Year: 2016
DOI identifier: 10.1214/15-AOP1001
OAI identifier: oai:arXiv.org:1103.5259
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