The Euclidean Matching Problem

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Springer, Oct 24, 2016 - Science - 136 pages
This 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 Introduction
1
2 Graphs and Optimization
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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About the author (2016)

Gabriele Sicuro is a postdoctoral researcher at the Centro Brasileiro de Pesquisas Físicas, in Rio de Janeiro. Born in 1987, he obtained his master degree in Physics at University of Salento, in Lecce, in 2011 and then his doctorate in Physics at University of Pisa in January 2015.

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