The spectral function of random resistor networks

The effective complex conductivity σ_eff of a two-component material can be conveniently expressed as an integral transformation of a spectral function. The spectral function depends only on the geometry of the material, and can be used to calculate σ_eff for any particular choice of component conductivities. This is a very useful feature if the component conductivities can be varied (by changing the temperature or frequency, for example) at a fixed geometry. We present a derivation of the spectral function that identifies it as a density of states. We have made direct numerical calculations of the spectral function of two-dimensional random resistor networks. Two-dimensional discrete resistor networks are ideal for this study, as the Y-Δ transformation can be used as an algorithm to obtain the most detailed results to date. We identify the structure in the spectral function with clusters in the network. We give analytic expressions for the first five moments of the spectral function, which are identified as the expansion coefficients of the effective conductivity in weak-scattering theory, and compare these expressions with the moments calculated from the simulations.