Minimal spanning trees at the percolation threshold: a numerical calculation

TL;DR

This paper estimates the fractal dimension of minimal spanning trees on percolation clusters across dimensions up to five, introducing a robust analysis method for correlated fractal data and confirming theoretical predictions.

Contribution

Develops a new robust analysis technique for correlated fractal structures and applies it to minimal spanning trees on percolation clusters, confirming theoretical predictions.

Findings

Fractal dimension $d_s$ agrees with perturbation theory predictions.

New combined Prim-Kruskal algorithm reduces memory usage.

Method applicable to various randomly generated fractal structures.

Abstract

The fractal dimension of minimal spanning trees on percolation clusters is estimated for dimensions $d$ up to $d=5$. A robust analysis technique is developed for correlated data, as seen in such trees. This should be a robust method suitable for analyzing a wide array of randomly generated fractal structures. The trees analyzed using these techniques are built using a combination of Prim's and Kruskal's algorithms for finding minimal spanning trees. This combination reduces memory usage and allows for simulation of larger systems than would otherwise be possible. The path length fractal dimension $d_{s}$ of MSTs on critical percolation clusters is found to be compatible with the predictions of the perturbation expansion developed by T.S.Jackson and N.Read [T.S.Jackson and N.Read, Phys.\ Rev.\ E \textbf{81}, 021131 (2010)].