Finer estimates on the 2-dimensional matching problem
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 737-765.

We study the asymptotic behaviour of the expected cost of the random matching problem on a 2-dimensional compact manifold, improving in several aspects the results of [AST18]. In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2-dimensional closed manifold. We also sharpen the estimate of the error term given in [Led18] for the semi-discrete matching. As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.

Nous étudions le comportement asymptotique de l’espérance du coût du problème de couplage aléatoire dans une variété compacte de dimension 2, améliorant en de nombreux points les résultats de [AST18]. En particulier, nous simplifions la preuve originale et nous déterminons le coefficient dominant du développement asymptotique de l’espérance du coût du couplage aléatoire biparti dans une variété compacte de dimension 2. Nous précisons aussi l’estimation du terme d’erreur dans [Led18] pour le couplage semi-discret. Nous développons une estimation de contraction plus fine pour le flot de la chaleur sur des données aléatoires qui peut avoir un intérêt indépendant du reste de l’article.

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DOI: 10.5802/jep.105
Classification: 60D05,  49J55,  60H15
Keywords: Optimal transport, matching problem, heat semigroup
Ambrosio, Luigi 1; Glaudo, Federico 2

1 Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa, Italy
2 ETH Zürich Rämistrasse 101, 8092 Zürich, Switzerland
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Ambrosio, Luigi; Glaudo, Federico. Finer estimates on the $2$-dimensional matching problem. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 737-765. doi : 10.5802/jep.105. http://www.numdam.org/articles/10.5802/jep.105/

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