Minimal matchings of point processes

• Published in: Probability theory and related fields Vol. 184; no. 1-2; pp. 571 - 611
• Main Authors: Holroyd, Alexander E., Janson, Svante, Wästlund, Johan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2022; Springer Nature B.V
• Subjects: Altruism; Economics; Finance; Insurance; Management; Matching; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Parameters; Point process; Poisson process; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Questions; Stationary process; Statistics for Business; Theoretical; Matching; 60D05; 60G55; Point process; Stationary process; 05C70; Poisson process
• DOI: 10.1007/s00440-022-01151-y
• ISSN: 0178-8051; 1432-2064

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 → - ∞ is equivalent to Gale-Shapley stable matching. We also consider limits as γ approaches 0, 1 - , 1 + and ∞ . 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.