Effective medium theory for resistor networks in checkerboard geometries

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Abstract
The authors have considered the effective resistance of resistor networks which can be mapped onto a checkerboard geometry. Each square in the board is represented by four resistors of the same magnitude, Ri, where Ri=R1 or R2, with probabilities p1 and p2=1-p1. The configuration of the four resistors in a square can be chosen naturally in four different ways. For each of these they have calculated the effective medium theory (EMT) result for the effective resistance and compared it with the result of a numerical calculation for a large random network of the appropriate configuration. The agreement between EMT and our simulation is very good. It is worth noticing that the effective resistance falls outside the corresponding Hashin-Shtrikman bounds to the effective resistance of a continuous two-phase material. From EMT the authors have obtained percolation thresholds, which contain transcendental numbers (e.g. 1/ pi ).