Multicolour Poisson Matching

TL;DR

This paper investigates translation-invariant schemes for partitioning multiple coloured Poisson point processes into allowed family types, characterizing existence conditions and tail behaviour of family diameters across different regimes.

Contribution

It generalizes previous matching schemes to multiple colours and family types, providing a comprehensive characterization of existence and tail behaviour in various regimes.

Findings

Characterized when translation-invariant schemes exist for multicolour Poisson processes.

Identified two regimes for tail behaviour of family diameters based on intensity vectors.

Determined optimal tail behaviour for deterministic partitions in one dimension.

Abstract

Consider several independent Poisson point processes on R^d, each with a different colour and perhaps a different intensity, and suppose we are given a set of allowed family types, each of which is a multiset of colours such as red-blue or red-red-green. We study translation-invariant schemes for partitioning the points into families of allowed types. This generalizes the 1-colour and 2-colour matching schemes studied previously (where the sets of allowed family types are the singletons {red-red} and {red-blue} respectively). We characterize when such a scheme exists, as well as the optimal tail behaviour of a typical family diameter. The latter has two different regimes that are analogous to the 1-colour and 2-colour cases, and correspond to the intensity vector lying in the interior and boundary of the existence region respectively. We also address the effect of requiring the partition to be a deterministic function (i.e. a factor) of the points. Here we find the optimal tail behaviour in dimension 1. There is a further separation into two regimes, governed by algebraic properties of the allowed family types.