arXiv 1708.045192017-08math.PR

基于Poisson加权无限树的高维稳定匹配

Stable matchings in high dimensions via the Poisson-weighted infinite tree

Alexander E. Holroyd, James B. Martin, Yuval Peres

备注：: 29 pages

摘要：

考虑了非对称色约束下$\mathbb{R}^d$中两个独立Poisson过程的稳定匹配.蓝色点只能匹配红色点，而红色点可以匹配任何一种颜色的点。不知道是否存在红色和蓝色过程的强度的选择，在该选择下所有点都匹配。我们证明，对于任何固定的强度，有足够高的维度不匹配的蓝点。实际上，如果红色与蓝色强度的比率是$ρ$，则不匹配的蓝色点的强度收敛到$e^{-ρ}/(1+ρ)$，作为$d\to\infty$。我们还建立了类似的结果，某些多色的变种。我们的证明使用稳定匹配的泊松加权无限树（PWIT），这可以通过微分方程进行分析。PWIT已被用于在许多设置中作为一个缩放限制的模型，涉及完全图与独立的边权重，但我们相信，这是第一个严格的应用程序，高维欧几里得空间。最后，我们分析了层次度量下的非对称匹配问题，并表明存在不匹配点的所有强度。

英文摘要：

We consider the stable matching of two independent Poisson processes in $\mathbb{R}^d$ under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown whether there exists a choice of intensities of the red and blue processes under which all points are matched. We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension. Indeed, if the ratio of red to blue intensities is $ρ$ then the intensity of unmatched blue points converges to $e^{-ρ}/(1+ρ)$ as $d\to\infty$. We also establish analogous results for certain multi-color variants. Our proof uses stable matching on the Poisson-weighted infinite tree (PWIT), which can be analyzed via differential equations. The PWIT has been used in many settings as a scaling limit for models involving complete graphs with independent edge weights, but we believe that this is the first rigorous application to high-dimensional Euclidean space. Finally, we analyze the asymmetric matching problem under a hierarchical metric, and show that there are unmatched points for all intensities.

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