Exact result for the effective conductivity of a continuum percolation model

A random two-dimensional checkerboard of squares of conductivities 1 and 8 in proportions p and 1 - p is considered. Classical duality implies that the effective conductivity obeys o* = V8 at p = 1/2. It is rigorously found here that to leading order as 8--0, this exact result holds for all p in the interval (1- pc,pc), where pc=0.59 is the site percolation probability, not just at p = 1/2. In particular, o*(p,8)=78+O (8), as 8 -- 0. which is argued to hold for complex 8 as well. The analysis is based on the identification of a "symmetric" backbone, which is statistically invariant under interchange of the components for any pE(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.