One-loop diagrams in the Random Euclidean Matching Problem

TL;DR

This paper analyzes the Euclidean matching problem using diagrammatic expansions, revealing a vanishing mass at zero momentum and contrasting it with non-Euclidean cases where anomalous scaling occurs.

Contribution

It introduces a diagrammatic approach to analyze one-loop corrections in the Euclidean matching problem, providing new insights into its asymptotic behavior.

Findings

Vanishing of the mass at zero momentum in the Euclidean case

Explicit diagrammatic rules for the expansion

Numerical verification of anomalous scaling in non-Euclidean case

Abstract

The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e. the optimal matching problem between points randomly distributed on a $d$-dimensional Euclidean space, where the cost to minimize depends on the points' pairwise distances. Using Mayer's cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation. We give the diagrammatic rules for each term of the expansion, and we analyze in detail the one-loop diagrams. A characteristic feature of the theory, when diagrams are perturbatively computed around the mean field part of the action, is the vanishing of the mass at zero momentum. In the non-Euclidean case of uncorrelated costs instead, we predict and numerically verify an anomalous scaling for the sub-sub-leading correction to the asymptotic average cost.