论文标题
点过程的最小匹配
Minimal matchings of point processes
论文作者
论文摘要
假设在$ r^d $中，红点和蓝点形成了同等强度的独立均匀泊松过程。对于正面（分别为负）参数$γ$，我们考虑的红蓝色匹配度以局部最小化（分别为最大化）边缘长度的$γ$ th功率的总和，可能会局部最小化无与伦比的点的数量。该参数可以看作是公平度的衡量标准。极限$γ\ to- \ \ infty $等效于Gale-Shapley稳定匹配。我们还将限制视为$γ$接近$ 0 $，$ 1- $，$ 1+$和$ \ infty $。我们专注于尺寸$ d = 1 $。我们证明，几乎没有这样的匹配能力是无与伦比的。 （此问题以更高的$ D $开放）。对于每个$γ<1 $ $，我们确定几乎存在这种唯一的匹配，并且可以表示为要点的限制因素。此外，当且仅当$ r <1/2 $时，其典型的边缘长度具有有限的$ r $第thormth。相比之下，对于$γ= 1 $，有很多匹配，而对于$γ> 1 $，有很多人可以数很多，但是不可能以翻译不变的方式选择一个。我们获得的存在会导致更高的维度（涵盖许多但不是所有情况）。我们还解决了单色比赛的类似问题。
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.
