Scaling hypothesis for the Euclidean bipartite matching problem. II. Correlation functions

Area of Science:

• Statistical Physics
• Computational Mathematics
• Geometric Probability

Background:

• The random Euclidean bipartite matching problem involves pairing points in a space to minimize total distance.
• Understanding optimal matching is crucial in various fields, including physics and computer science.
• Previous work has focused on average costs, but correlation functions offer deeper insights.

Purpose of the Study:

• To analyze the two-point correlation function for optimal matching in the d-dimensional hypertorus setting.
• To investigate both grid-Poisson and Poisson-Poisson matching scenarios.
• To establish a relationship between the correlation function and the Green's function of the Laplace operator.

Main Methods:

• Utilizing an ansatz by Caracciolo et al. for evaluating average optimal matching costs.
• Applying techniques from statistical physics and probability theory.
• Deriving the two-point correlation function analytically.

Main Results:

• The two-point correlation function for optimal matching was derived.
• A strict relationship was established between this correlation function and the Green's function of the Laplace operator on the hypertorus.
• The analysis covered both grid-Poisson and Poisson-Poisson matching problems.

Conclusions:

• The derived correlation function provides a more detailed characterization of optimal matchings than average costs alone.
• The connection to the Laplace operator's Green's function offers a new perspective and potential for further theoretical development.
• This work advances the understanding of random matching problems in high-dimensional spaces.