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 ε=6-d for d\leq 6 using the connection with critical percolation, with the result D_p = 2 - ε/7 + O(ε^2).

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

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Cited by in corpus (7)

• Theory of minimum spanning trees I: Mean-field theory and strongly disordered spin-glass model
• Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane
• Clustering with minimum spanning trees: How good can it be?
• Fractal dimension of interfaces in Edwards-Anderson spin glasses for up to six space dimensions
• Minimal spanning trees at the percolation threshold: a numerical calculation
• Loop erased random walk on percolation cluster: Crossover from Euclidean to fractal geometry
• Large Deviation Properties of Minimum Spanning Trees for Random Graphs