## The Euclidean Matching ProblemThis thesis discusses the random Euclidean bipartite matching problem, i.e., the matching problem between two different sets of points randomly generated on the Euclidean domain. The presence of both randomness and Euclidean constraints makes the study of the average properties of the solution highly relevant. The thesis reviews a number of known results about both matching problems and Euclidean matching problems. It then goes on to provide a complete and general solution for the one dimensional problem in the case of convex cost functionals and, moreover, discusses a potential approach to the average optimal matching cost and its finite size corrections in the quadratic case. The correlation functions of the optimal matching map in the thermodynamical limit are also analyzed. Lastly, using a functional approach, the thesis puts forward a general recipe for the computation of the correlation function of the optimal matching in any dimension and in a generic domain. |

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### Contents

1 | |

5 | |

3 Random Optimization Problems and Statistical Mechanics | 25 |

4 Euclidean Matching Problems | 58 |

5 Conclusions and Perspectives | 119 |

Appendix A Additional Classical Results on Glasses | 121 |

Appendix B The Wiener Process and the Brownian Bridge Process | 129 |

Curriculum Vitae | 134 |

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Aldous ansatz assignment problem average optimal cost belief propagation Bethe lattice bipartite matching bipartite matching problem boundary conditions Brownian bridge Brownian bridge process called Caracciolo cavity method combinatorial complete bipartite graph complete graph computation convex correlation function corresponding cost function cycles defined denote dimension dimensional disordered systems edge Euclidean bipartite matching Euclidean matching problem factor graph finite size corrections given graph G Graph(V,U;E Graph(V;E Hamiltonian Hungarian algorithm introduced Ising model Let us consider limit matrix mean field Mézard and Parisi obtained optimal map optimal matching optimal transport particular perfect matching Phys previous expression probability distribution density random assignment problem random Euclidean random optimization random optimization problems rbmp replica symmetric rmmp Sicuro SK-model spin glass statistical physics subgraphs suppose theorem theory tree vertex vertices weight Wiener process