L. Berlyand et K. Golden, EXACT RESULT FOR THE EFFECTIVE CONDUCTIVITY OF A CONTINUUM PERCOLATION MODEL, Physical review. B, Condensed matter, 50(4), 1994, pp. 2114-2117
A random two-dimensional checkerboard of squares of conductivities 1 a
nd delta in proportions p and 1-p is considered. Classical duality imp
lies 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 probabilit
y, not just at p = 1/2. In particular, sigma(p,delta) = square-root d
elta + O(delta), as delta --> 0, which is argued to hold for complex d
elta as well. The analysis is based on the identification of a ''symme
tric'' backbone, which is statistically invariant under interchange of
the components for any p is-an-element-of (1-p(c),p(c)), like the ent
ire checkerboard at p = 1/2. This backbone is defined in terms of ''ch
oke points'' for the current, which have been observed in an experimen
t.