A Poisson allocation of optimal tail

  title={A Poisson allocation of optimal tail},
  author={Roland Mark'o and 'Ad'am Tim'ar},
  journal={arXiv: Probability},
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… 

Figures from this paper

Poisson allocations with bounded connected cells
Given a homogenous Poisson point process in the plane, we prove that it is possible to partition the plane into bounded connected cells of equal volume, in a translation-invariant way, with each
Optimal transport between random measures
We study couplings $q^\bullet$ of two equivariant random measures $\lambda^\bullet$ and $\mu^\bullet$ on a Riemannian manifold $(M,d,m)$. Given a cost function we ask for minimizers of the mean
Dynamic Spatial Matching
It is shown that one can achieve nearly the same cost under the semi-dynamic model as under the static model, despite uncertainty about the future, and that, under these two models, d=1 is the only case where cost far exceeds the expected distance to the nearest neighboring supply unit.
There is no stationary cyclically monotone Poisson matching in 2d
We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension d = 2. The proof combines the harmonic approximation result from [10] with local
A factor matching of optimal tail between Poisson processes
Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e.,


A Stable Marriage of Poisson and Lebesgue
Let be a discrete set in R d . Call the elements of centers. The well-known Voronoi tessellation partitions R d into polyhedral regions (of varying sizes) by allocating each site of R d to the
Extra heads and invariant allocations
Let n be an ergodic simple point process on E d and let n* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Π and n*; that is, one can
How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields
We consider the following problem: given an i.i.d. family of Bernoulli random variables indexed by Z d , find a random occupied site X ∈ Z d such that relative to X, the other random variables are
Phase Transitions in Gravitational Allocation
Given a Poisson point process of unit masses (“stars”) in dimension d ≥ 3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. In earlier work, we showed the
Poisson Matching
Suppose that red and blue points occur as independent homogeneous Poisson processes in Rd. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For
Gravitational allocation to Poisson points
For d ≥ 3, we construct a non-randomized, fair and translationequivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R d , defined by allocating to each of
Connected allocation to Poisson points in $\mathbb{R}^2$
This note answers one question in [1] concerning the connected allocation for the Poisson process in $\mathbb{R}^2$. The proposed solution makes use of the Riemann map from the plane minus the
Optimal transport from Lebesgue to Poisson
This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean
Connected allocation to Poisson points in R^2
This note answers one question in [math.PR/0505668], concerning the connected allocation for the Poisson process in R^2. The proposed solution makes use of the Riemann map from the plane minus the
On optimal matchings
Givenn random red points on the unit square, the transportation cost between them is tipically √n logn, where logn is the number of red points in the ellipsoidal plane.