2016/05/20 by Amir, Gideon, Angel, Omer, Holroyd, Alexander E.
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1605.06485
Consider several independent Poisson point processes on Rd, 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.