Dependence of conductance on percolation backbone mass

We study [Formula Presented] the average conductance of the backbone, defined by two points separated by Euclidean distance r, of mass [Formula Presented] on two-dimensional percolation clusters at the percolation threshold. We find that with increasing [Formula Presented] and for fixed [Formula Presented] 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 [Formula Presented] We explain this behavior by studying the distribution of shortest paths between the two points on clusters with a given [Formula Presented] We also study the dependence of conductance on [Formula Presented] above the percolation threshold and find that (i) slightly above [Formula Presented] the conductance first decreases and then increases with increasing [Formula Presented] and (ii) further above [Formula Presented] the conductance increases monotonically for all values of [Formula Presented] as is the case for homogeneous systems.