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

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

2009-09-29

arxiv.org/abs/0909.5343v1

Phys. Rev. E 81, 021131 (2010)

Physics

Condensed Matter

Disordered Systems and Neural Networks

33 pages, 5 figures, submitted to PRE; part I available at arXiv:0902.3651

Scientific paper

10.1103/PhysRevE.81.021131

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).

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