Theory of minimum spanning trees II: exact graphical methods and perturbation expansion at the percolation threshold
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).
Keywords
Cite
@article{arxiv.0909.5343,
title = {Theory of minimum spanning trees II: exact graphical methods and perturbation expansion at the percolation threshold},
author = {T. S. Jackson and N. Read},
journal= {arXiv preprint arXiv:0909.5343},
year = {2010}
}
Comments
33 pages, 5 figures, submitted to PRE; part I available at arXiv:0902.3651