EXACT RESULT FOR THE EFFECTIVE CONDUCTIVITY OF A CONTINUUM PERCOLATION MODEL

A random two-dimensional checkerboard of squares of conductivities 1 and delta in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys sigma* = square-root delta at p = 1/2. It is rigorously found here that to leading order as delta --> 0, this exact result holds for all p in the interval (1-p(c),p(c)), where p(c) almost-equal-to 0. 59 is the site percolation probability, not just at p = 1/2. In particular, sigma*(p,delta) = square-root delta + O(delta), as delta --> 0, which is argued to hold for complex delta as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p is-an-element-of (1-p(c),p(c)), like the entire checkerboard at p = 1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.