Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold

Journal Article · Mon Feb 15 04:00:00 EST 2010 · Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print)

DOI:https://doi.org/10.1103/PHYSREVE.81.021131· OSTI ID:21344708

Jackson, T S; Read, N ^[1]

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1. Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120 (United States)

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{sub p} of paths on the latter MST; our previous results lead us to predict that D{sub p}=2 for d>d{sub c}=6. Using a renormalization-group approach, we confirm the result for d>6 and calculate D{sub p} to first order in epsilon=6-d for d<6 using the connection with critical percolation, with the result D{sub p}=2-epsilon/7+O(epsilon{sup 2}).

• OSTI ID: 21344708
• Journal Information: Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print), Journal Name: Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print) Journal Issue: 2 Vol. 81; ISSN 1539-3755
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
FRACTALS
MATHEMATICAL SPACE
PERTURBATION THEORY
RANDOMNESS
RENORMALIZATION
SPACE