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Poisson matching. (English) Zbl 1175.60012
Authors’ abstract: Suppose that red and blue points occur as independent Poisson processes in \(\mathbb{R}^d\). We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions \(d=1,2\), the matching distance \(X\) from a typical point to its partner must have infinite \(d/2\)th moment, while in dimension \(d\geq3\) there exist schemes where \(X\) has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance \(X\) for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in \(d=1\), but far from optimal in \(d\geq3\). For the problem of matching Poisson points of a single color to each other, in \(d=1\) there exist schemes where \(X\) has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process, then \(X\) must have infinite mean.

60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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