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

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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 Dₚ of paths on the latter MST; our previous results lead us to predict that Dₚ=2 for d>dc=6. Using a renormalization-group approach, we confirm the result for d>6, and calculate Dₚ to first order in ε=6-d for d≤ 6 using the connection with critical percolation, with the result Dₚ = 2 - ε/7 + O(ε²).