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

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<6 using the connection with critical percolation, with the result D(p)=2-epsilon/7+O(epsilon(2)) .