Dependence of Conductance on Percolation Backbone Mass

TL;DR

This study investigates how the average conductance of the backbone in two-dimensional percolation clusters depends on the backbone's mass and separation distance, revealing a decrease to a constant with increasing mass, contrasting with homogeneous systems.

Contribution

It provides new insights into conductance behavior on percolation backbones, highlighting differences from non-random fractals and explaining the underlying path distributions.

Findings

Conductance decreases to a constant with increasing backbone mass at fixed distance.

Behavior differs from homogeneous systems where conductance increases with size.

Analysis of shortest path distributions explains the conductance trend.

Abstract

On two-dimensional percolation clusters at the percolation threshold, we study $<\sigma(M_B,r)>$, the average conductance of the backbone, defined by two points separated by Euclidean distance $r$, of mass $M_B$. We find that with increasing $M_B$ and for fixed r, $<\sigma(M_B,r)>$ asymptotically {\it decreases} to a constant, in contrast with the behavior of homogeneous sytems and non-random 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 $M_B$ slightly above the percolation threshold.