## 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.

Original language | English |
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Pages (from-to) | 1811-1833 |

Number of pages | 23 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 57 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2021 |

### Bibliographical note

Funding Information:We wish to thank the anonymous referees for their useful comments. This work was initiated during a UBC Probability Summer School, and advanced while some of the authors were visiting Microsoft Research. We are grateful to Microsoft Research for their support. GA is supported by the ISF and GIF. OA is supported by NSERC.

Publisher Copyright:

© Association des Publications de l’Institut Henri Poincaré, 2021

## Keywords

- Factor map
- Invariant matching
- Invariant partition
- Point process
- Poisson process