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 $D_{\mathrm{p}}$ of paths on the latter MST; our previous results lead us to predict that $D_{\mathrm{p}}=2$ for $d>d_c=6$. Using a renormalization-group approach, we confirm the result for $d>6$ and calculate $D_{\mathrm{p}}$ to first order in $\epsilon =6-d$ for $d<6$ using the connection with critical percolation, with the result $D_{\mathrm{p}}=2-\epsilon ∕7+O\left({\epsilon }^2\right)$.

• Received 29 September 2009

DOI:https://doi.org/10.1103/PhysRevE.81.021131