ETD

• concentration bounds
• Hopf-Lax semigroup
• optimal transport
• p-Laplacian
• pde
• random matching

This thesis investigates the asymptotic behavior of the random bipartite matching problem on the two-dimensional square. Given two sets of independent random points uniformly distributed on the domain, we study the properties of the p-Wasserstein distance between the empirical measures they induce for a general cost with exponent p>1, as the number of points tends to infinity.
In particular, inspired by a PDE ansatz first introduced by Caracciolo et al. in the physics literature and later rigorously proved by Ambrosio et al., we extend from the two-dimensional torus to the two-dimensional square a remarkable asymptotic connection between the expected value of the Wasserstein cost and the average energy of a nonlinear PDE of q-Poisson type, where q is the dual exponent of p. This PDE naturally arises from a formal linearization of the Monge-Ampère equation. Unlike the periodic case of the torus, the presence of a boundary introduces new analytical challenges, which arise in the study of the associated Neumann problem. A key step to overcome these difficulties relies on a symmetrization strategy, which allows us to bypass boundary issues and apply techniques analogous to those valid for the torus. This enables us to establish a cornerstone deterministic bound linking the two quantities, upon which the entire probabilistic proof is built.
Finally, we use this framework to establish a quantitative concentration result for the Wasserstein cost in the semi-discrete setting. This is achieved by proving concentration for the energy of the associated PDE, a step that ultimately relies on results from the nonlinear Calderón-Zygmund theory introduced by Iwaniec. These results, although still dependent on the peculiar symmetry properties of the square, suggest the possibility of extending the asymptotic analysis of random matchings to more general manifolds with boundary, and also represent a preliminary step towards understanding the open problem of the existence of the limit of the cost for a general exponent p>1.