Tail Bounds for the Stable Marriage of Poisson and Lebesgue

Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval

Tail Bounds for the Stable Marriage of Poisson and Lebesgue

Authors

Hoffman, Christopher

Holroyd, Alexander E.

Peres, Yuval

Description

Let Ξbe a discrete set in R^d. Call the elements of Ξcenters. The well-known Voronoi tessellation partitions R^d into polyhedral regions (of varying volumes) by allocating each site of R^d to the closest center. Here we study allocations of R^d to Ξin which each center attempts to claim a region of equal volume α. We focus on the case where Ξarises from a Poisson process of unit intensity. It was proved in math.PR/0505668 that there is a unique allocation which is stable in the sense of the Gale-Shapley marriage problem. We study the distance X from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite α. In the critical case α=1 we prove a power law upper bound on X in dimension d=1. It is an open problem to prove any upper bound in d\geq 2. (Power law lower bounds were proved in math.PR/0505668 for all d). In the non-critical cases α<1 and α>1 we prove exponential upper bounds on X.
30 pages, 2 figures