Contributions to the optimal transport problem and its regularity

This thesis deals with the optimal transport problem, in particular with regularity properties shared by optimal transport maps. The first part of this manuscript provides a new proof of the Caffarelli contraction theorem, stating that the optimal transport map from the gaussian measure to a measure with a uniformly log-concave density with respect to the gaussian measure is 1-Lipschitz. The strategy developped here differs from the previous ones in the fact that it directly exploits the minimization problem, through an entropic regularization of optimal transport and a variational characterization of the Lipschitz property of an optimal transport map proved by Gozlan and Juillet. The second part extends the variational approach, introduced by Goldman and Otto for the regularity theory of optimal transport maps, to the situation of general cost functions. The main contribution is the introduction, in the context of optimal transport, of the notion of almost-minimality, which is already well established in minimal surfaces. We show that an optimal transport plan for a cost function close to being quadratic is almost-optimal for the quadratic cost, allowing us to solve the issue raised by the non-existence of a Benamou-Brenier equivalence for a general cost function. In the third part, we obtain the optimal rate of convergence, in W1 and W2 Wasserstein distance, of the empirical spectral measure of Ginibre matrices to the circular law. This relies on an estimation of the fluctuations of the number of points in a domain, using the determinantal structure of the point process, and an iterative construction of intermediate measures, from microscopic to macroscopic scales, which allows to link the W2 distance to the variance of the number of points.