Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - F. Glaudo

Finer estimates on the 2-dimensional matching problem

created by glaudo on 16 Oct 2018


Submitted Paper

Inserted: 16 oct 2018
Last Updated: 16 oct 2018

Pages: 23
Year: 2018


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 L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2-dimensional matching problem). 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 by M. Ledoux (On optimal matching of Gaussian samples II) 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.


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