Vol. 2, No. 2, 2021

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Convergence of asymptotic costs for random Euclidean matching problems

Michael Goldman and Dario Trevisan

Vol. 2 (2021), No. 2, 341–362
Abstract

We investigate the average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d 3 dimensions, where the matching cost between two points is given by any power p 1 of their Euclidean distance. As n grows, we prove convergence, after a suitable renormalization, towards a finite and positive constant. We also consider the analogous problem of optimal transport between n points and the uniform measure. The proofs combine subadditivity inequalities with a PDE ansatz similar to the one proposed in the context of the matching problem in two dimensions and later extended to obtain upper bounds in higher dimensions.

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Keywords
matching problem, optimal transport, geometric probability
Mathematical Subject Classification
Primary: 35J05, 39B62, 60D05, 60F25, 90C05
Milestones
Received: 15 September 2020
Accepted: 3 December 2020
Published: 22 May 2021
Authors
Michael Goldman
Laboratoire Jacques-Louis Lions
Université de Paris, CNRS
Sorbonne-Université
Paris
France
Dario Trevisan
Dipartimento di Matematica
Università di Pisa
Pisa
Italy