Duality and effective conductivity of random two-phase flat systems

The possible functional forms of the effective conductivity $\sigma_e$ of the randomly inhomogeneous two-phase systems at arbitrary values of concentrations are discussed. Two explicit approximate expressions for effective conductivity are found using a duality relation, a series expansion of $\sigma_e$ in the inhomogeneity parameter $z$ and some additional conjectures about functional form of $\sigma_e$. They differ from the effective medium approximation, satisfy all necessary requirements and reproduce the known formulas for $\sigma_e$ in weakly inhomogeneous case. This can signify also that $\sigma_e$ of the two-phase randomly inhomogeneous systems may be a nonuniversal function, depending on some details of the structure of the random inhomogeneities.