Mean field and corrections for the Euclidean Minimum Matching problem

TL;DR

This paper estimates the Euclidean minimum matching length in various dimensions, introduces a correlation-ignoring approximation, and improves it with link correlations, providing precise results and insights into high-dimensional behavior.

Contribution

It provides accurate estimates of the Euclidean minimum matching constant across dimensions and develops a correlation-based approximation method with improved accuracy.

Findings

Precise estimates of eta_{MM}^E(d) for 2 0.

Correlation-ignoring model closely approximates eta_{MM}^E(d) for d .

Including three-link correlations reduces error to 0.4% at d=2.

Abstract

Consider the length $L_{MM}^E$ of the minimum matching of N points in d-dimensional Euclidean space. Using numerical simulations and the finite size scaling law $< L_{MM}^E > = \beta_{MM}^E(d) N^{1-1/d}(1+A/N+... )$, we obtain precise estimates of $\beta_{MM}^E(d)$ for $2 \le d \le 10$. We then consider the approximation where distance correlations are neglected. This model is solvable and gives at $d \ge 2$ an excellent ``random link'' approximation to $\beta_{MM}^E(d)$. Incorporation of three-link correlations further improves the accuracy, leading to a relative error of 0.4% at d=2 and 3. Finally, the large d behavior of this expansion in link correlations is discussed.