The Euclidean Matching Problem

In the present thesis we discuss the so-called Euclidean Matching Problem. We overview the main results obtained in the last fifty years on Random Optimization Problems, stressing the important contributions given by Statistical Physics and describing the most powerful tools used, prevalently borrowed from Spin Glass Theory. In the spirit of the classic results, we studied the Euclidean Matching Problem following very different approaches, showing its connections with the Theory of Stochastic Processes, Measure Theory and the Theory of Disordered Systems. We developed new methods to analyse this combinatorial problem inspired by its continuum versions as transport problem between measures and treating Euclidean correlations as a perturbation to the purely random case.