# A PDE approach to a 2-dimensional matching problem

created by trevisan on 15 Nov 2016
modified on 13 Dec 2016

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Preprint

Inserted: 15 nov 2016
Last Updated: 13 dec 2016

Year: 2016

Abstract:

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by S. Caracciolo et al. (Phys. Rev. E, {\bf 90} 012118, 2014) that linearise'' the Monge-Ampère equation.