Spectral functions for two-dimensional random conductors

We collect together a number of results for the spectral function for two-dimensional random conductors. Exact solutions are available for the conductivity of sheets containing a few elliptical inclusions. We show that exact results can also be obtained for polygonal holes using conformal mapping techniques. For polygonal inclusions with arbitrary conductivity, the conductivity can be expressed in terms of a spectral function which we have obtained numerically using a boundary element method. In this method the singularities associated with the sharp corners are explicitly taken care of. Spectral functions can also be used to obtain an exact solution for the conductivity when the effect of pairs of circular inclusions is also included. Throughout this paper we express the conductivity of the composite in terms of the total induced dipole moment and exploit the simplifications due to the reciprocity theorem in two dimensions.