On universality of conductivity of planar random self-dual systems

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General properties of the effective conductivity sigmaₑ of planar isotropic randomly inhomogeneous two-phase self-dual systems are investigated. A new approach for finding out sigmaₑ of random systems based on a duality, a series expansion in the inhomogeneous parameter z and additional assumptions, is proposed. Two new approximate expressions for sigmaₑ at arbitrary values of phase concentrations are found. They satisfy all necessary inequalities, symmetries, including a dual one, and reproduce known results in various limiting cases. Two corresponding models with different inhomogeneity structures, whose sigmaₑ coincide with these expressions, are constructed. First model describes systems with a finite maximal characteristic scale of the inhomogeneities. In this model sigmaₑ is a solution of the approximate functional equation, generalizing the duality relation. The second model is constructed from squares with random layered structure. The difference of sigmaₑ for these models means a nonuniversality of the effective conductivity even for binary random self-dual systems. The first explicit expression for sigmaₑ can be used also for approximate description of various inhomogeneous systems with compact inclusions of the second phase. The percolation problem of these models is briefly discussed.