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Area of Science:

  • Combinatorics
  • Probability Theory
  • Statistical Physics

Background:

  • The Euclidean bipartite matching problem involves finding optimal pairings between points.
  • Brownian bridge processes model random paths constrained between two points.

Purpose of the Study:

  • To explore the relationship between Euclidean bipartite matching and Brownian bridge processes.
  • To leverage this equivalence for solving combinatorial optimization problems.

Main Methods:

  • Establishing an equivalence relation between matching on a line/circumference and Brownian bridge processes.
  • Utilizing probabilistic methods to analyze the thermodynamic limit.

Main Results:

  • The equivalence allows for the computation of correlation functions and optimal costs in the thermodynamic limit.
  • The minimax problem for bipartite matching on the line and circumference is solved.

Conclusions:

  • The established equivalence provides a powerful tool for analyzing combinatorial problems using probabilistic methods.
  • Properties of average cost and correlation functions are characterized.