Minimal spanning trees at the percolation threshold: a numerical calculation

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)].