Diagram technique for perturbation theory calculations of the effective   conductivity of two-dimensional systems
  The perturbation theory for calculation of the effective conductivity of the plane consisting of pieces of different conductivities is constructed and the convenient diagram technique for this perturbation theory is elaborated. It is shown that for the chessboard perturbative calculations give results which are in agreement with the well-known formula $\sigma_{eff} = \sqrt{\sigma_1\sigma_2}$. The components of the tensor of effective conductivity for the anisotropic three-color chessboard are calculated. It is shown that the isotropic (symmetric) part of effective conductivity calculated up to the sixth order of perturbation theory satisfies the Bruggeman effective medium equation for symmetric three-color structures with equally partitioned components. We also consider isotropic three-color chessboard with nonequal weights of colors. In this case the perturbation theory already in fourth order contradicts the results following from the Bruggeman equation for nonequal weights. 
I.M. Khalatnikov and A.Yu. Kamenshchik