A Poisson allocation of optimal tail

@article{Marko2011APA,
  title={A Poisson allocation of optimal tail},
  author={Roland Mark'o and 'Ad'am Tim'ar},
  journal={arXiv: Probability},
  year={2011}
}
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… 
View PDF on arXiv
5 Citations
Highly Influential Citations
1
Background Citations
1
Methods Citations
2

Figures from this paper

5 Citations

Citation Type

Citation Type

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.,…
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
TLDR
Across cases, this work finds that the platform can achieve cost nearly as low as the expected distance to the nearest neighboring supply unit is achievable, i.e., the matching constraint does not cause an increase in cost either.
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…
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…

References

SHOWING 1-10 OF 11 REFERENCES
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
TLDR
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.
...
...