Calculus of Variations and Geometric Measure Theory

L. Ambrosio - F. Stra - D. Trevisan

A PDE approach to a 2-dimensional matching problem

created by trevisan on 15 Nov 2016
modified on 10 Jun 2017



Inserted: 15 nov 2016
Last Updated: 10 jun 2017

Year: 2016

ArXiv: 1611.04960 PDF


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-Amp\`ere equation.

Keywords: Optimal transport, minimal matching, geometric probability