Minimal matchings of point processes

Abstract

Suppose that red and blue points form independent homogeneous Poisson processes of equal intensity in $R^d$. For a positive (respectively, negative) parameter $\gamma$ we consider red-blue matchings that locally minimize (respectively, maximize) the sum of $\gamma$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 $\gamma\to-\infty$ is equivalent to Gale-Shapley stable matching. We also consider limits as $\gamma$ 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 $\gamma<1$ 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 $\gamma=1$ there are uncountably many matchings, while for $\gamma>1$ 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.