Exact result for the effective conductivity of a continuum percolation model

A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ 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(1-pc,pc), 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.