The Euclidean Matching Problem

This work explores the random Euclidean bipartite matching problem, providing insights into optimal matching costs and correlation functions in various dimensions. The Euclidean Matching Problem is a key reference.

In The Euclidean Matching Problem, the author delves into the complexities of random Euclidean bipartite matching, which involves pairing points from two distinct sets generated within the Euclidean space. This exploration is crucial as it highlights how randomness interacts with geometric constraints, shaping the average properties of solutions. The thesis not only surveys existing literature on matching problems but also emphasizes the unique challenges posed by the Euclidean framework.

The work presents a comprehensive solution to the one-dimensional case, particularly focusing on convex cost functionals. Here, the author meticulously discusses the implications of these findings for average optimal matching costs, while also addressing finite size corrections, particularly in quadratic scenarios. This nuanced analysis provides valuable insights into the behavior of matching costs as the size of the point sets varies.

Furthermore, the thesis investigates the correlation functions of optimal matching maps within the thermodynamic limit. By employing a functional approach, the author proposes a generalized method for calculating these correlation functions across various dimensions and domains. This innovative perspective not only enriches the understanding of the matching problem but also opens avenues for future research in the field, making The Euclidean Matching Problem a significant contribution to the literature on combinatorial optimization.