  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.
