Dependence of conductance on percolation backbone mass

We study <sigma(M(B),r)>, the average conductance of the backbone, defined by two points separated by Euclidean distance r, of mass M(B) on two-dimensional percolation clusters at the percolation threshold. We find that with increasing M(B) and for fixed r, <sigma(M(B),r)> asymptotically decreases to a constant, in contrast with the behavior of homogeneous systems and nonrandom fractals (such as the Sierpinski gasket) in which conductance increases with increasing M(B). We explain this behavior by studying the distribution of shortest paths between the two points on clusters with a given M(B). We also study the dependence of conductance on M(B) above the percolation threshold and find that (i) slightly above p(c), the conductance first decreases and then increases with increasing M(B) and (ii) further above p(c), the conductance increases monotonically for all values of M(B), as is the case for homogeneous systems.