Gravitational allocation to Poisson points

• 成果类型: Article
• 署名作者: Chatterjee, Sourav; Peled, Ron; Peres, Yuval; Romik, Dan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.172.617
• 发表日期: 2010
• 页码: 617-671
• 关键词: Stable marriage; lebesgue; bounds; zeros

摘要

For d >= 3, we construct a non-randomized, fair, and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R-d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the allocation diameter, defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound P (X > R) <= C exp[ - cR(log R)(alpha d)] for all R > 2, where: alpha(d) = d-2/d for d >= 4; alpha(3) 3 can be taken as any number less than -4/3; and C and c are positive constants that depend on d and alpha(d). This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X > R).