Theory of minimum spanning trees II: exact graphical methods and perturbation expansion at the percolation threshold

TL;DR

This paper develops exact and perturbative methods for analyzing minimum spanning trees at the percolation threshold, confirming the fractal dimension of paths in high dimensions and calculating corrections below the critical dimension.

Contribution

It introduces an exact low-density expansion and a renormalizable perturbation expansion for MSTs on critical percolation clusters, extending understanding of their fractal properties.

Findings

Confirmed fractal dimension D_p=2 for d>6

Calculated D_p=2 - /7 + O(^2) for d

Developed a renormalization-group approach for MST analysis

Abstract

Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph, and use it to develop a continuum perturbation expansion for the MST on critical percolation clusters in space dimension d. The perturbation expansion is proved to be renormalizable in d=6 dimensions. We consider the fractal dimension D_p of paths on the latter MST; our previous results lead us to predict that D_p=2 for d>d_c=6. Using a renormalization-group approach, we confirm the result for d>6, and calculate D_p to first order in \epsilon=6-d for d\leq 6 using the connection with critical percolation, with the result D_p = 2 - \epsilon/7 + O(\epsilon^2).