Finer estimates on the $2$-dimensional matching problem

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 cite{Ambrosio-Stra-Trevisan2018}. 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 cite{Ledoux18} 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.