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A factor matching of optimal tail between Poisson processes

Adam Timar

Abstract

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., an equivariant measurable function of the point configurations), and with the property that the distance between a configuration point and its pair has a tail distribution that decays as fast as possible, namely, as $b\exp (-cr^d)$ with suitable constants $b,c>0$. Our proof relies on two earlier results: an allocation rule of similar tail for a Poisson point process, and a recent theorem that enables one to obtain perfect matchings from fractional perfect matchings in our setup.

A factor matching of optimal tail between Poisson processes

Abstract

Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., an equivariant measurable function of the point configurations), and with the property that the distance between a configuration point and its pair has a tail distribution that decays as fast as possible, namely, as with suitable constants . Our proof relies on two earlier results: an allocation rule of similar tail for a Poisson point process, and a recent theorem that enables one to obtain perfect matchings from fractional perfect matchings in our setup.

Paper Structure

This paper contains 3 theorems, 5 equations.

Key Result

Theorem 1

Let $\omega_1$ and $\omega_2$ be independent Poisson point processes of intensity 1 in the Euclidean space $\mathbb R^d$ of dimension $d\geq 3$. There exists a factor perfect matching $m$ between $\omega_1$ and $\omega_2$ with the property that with some $b,c>0$, where $m(0)$ is the matched pair of 0.

Theorems & Definitions (5)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Remark 4