arXiv 2012.071292020-12-13math.PR

点过程的极小匹配

Minimal matchings of point processes

Alexander E. Holroyd, Svante Janson, Johan Wästlund

摘要：

假设红点和蓝点构成$R^d$中独立的等强度齐次泊松过程。对于一个积极的（分别为负）参数$γ$，我们认为红蓝匹配，局部最小化（分别为最大化）$γ$ th权力的边缘长度的总和，局部最小化不匹配点的数量。该参数可以被视为公平性的衡量标准。极限$γ\to-\infty$等价于Gale-Shapley稳定匹配。我们还考虑了$γ$接近$0$、$1-$、$1+$和$\infty$的极限。我们关注维度$d=1$。我们证明，几乎肯定没有这样的匹配有不匹配的点。(This问题是开放的更高的$d$）。对于每一个$γ<1$，我们建立，几乎肯定有一个唯一的这样的匹配，它可以表示为一个有限的因素的点。而且，它的典型边长具有有限$r$阶矩当且仅当$r<1/2$ .相比之下，对于$γ=1$，有无数个匹配，而对于$γ>1$，有无数个匹配，但不可能以保留不变的方式选择一个。我们在更高的维度上得到了存在性结果（涵盖了许多但不是所有的情况）。我们解决类似的问题，单色匹配。

英文摘要：

Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in $R^d$. For a positive (respectively, negative) parameter $γ$ we consider red-blue matchings that locally minimize (respectively, maximize) the sum of $γ$th powers of the edge lengths, subject to locally minimizing the number of unmatched points. The parameter can be viewed as a measure of fairness. The limit $γ\to-\infty$ is equivalent to Gale-Shapley stable matching. We also consider limits as $γ$ approaches $0$, $1-$, $1+$ and $\infty$. We focus on dimension $d=1$. We prove that almost surely no such matching has unmatched points. (This question is open for higher $d$). For each $γ<1$ we establish that there is almost surely a unique such matching, and that it can be expressed as a finitary factor of the points. Moreover, its typical edge length has finite $r$th moment if and only if $r<1/2$. In contrast, for $γ=1$ there are uncountably many matchings, while for $γ>1$ there are countably many, but it is impossible to choose one in a translation-invariant way. We obtain existence results in higher dimensions (covering many but not all cases). We address analogous questions for one-colour matchings also.

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