The optimal matching problem is a classical random variational problem which may be interpreted as an optimal transport problem between two random discrete measures. Its easier instance deals with matching 2 $n$-clouds of i. i. d. uniformly distributed points. In recent years Caracciolo-Lucibello-Parisi-Sicuro made exact predictions on the convergence of the rescaled cost thanks to a first order linearization of the Monge-Amp\’ere equation. This approach was later justified by Ambrosio-Stra-Trevisan and quantitative bounds for the convergence of the proxies were later shown by Ambrosio-Glaudo-Trevisan. Such techniques have been repurposed by Benedetto-Caglioti to study the case of of i. i. d. random points with non-constant densities. By subadditivity and PDE arguments Ambrosio-Goldman-Trevisan were able to justify the latter for the convergence of the rescaled cost. We show annealed quantitative upper bounds for the approximating transport map in the case of i. i. d. points and weakly correlated points with non-constant densities. We extend our results to the case of unbalanced matching, i. e. matching between point clouds of different size and to point clouds sampled from a positive recurrent Markov chain.
Joint work with N. Clozeau (IST Austria).