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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.

##### MSC:
 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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