On homogenized conductivity and fractal structure in a high contrast continuum percolation model

In the previous article (Matsutani et al., 2012) we numerically investigated an electric potential problem with high contrast local conductivities (γ0γ0 and γ1γ1, 0<γ0≪γ10<γ0≪γ1) for a two-dimensional continuum percolation model (CPM). As numerical results, we showed there that the equipotential curves exhibit the fractal structure around the threshold pcpc and gave an approximated curve representing a relation between the homogenized conductivity and the volume fraction p   over [pc,1][pc,1]. In this article, using the duality of the conductivities and the quasi-harmonic properties, we re-investigate these topics to improve these results. We show that at γ0→0γ0→0, the quasi-harmonic potential problem in CPM is quasiconformally equivalent to a random slit problem, which leads us to an observation between the conformal property and the fractal structure at the threshold. Further we extend the domain [pc,1][pc,1] of the approximated curve to [0,1][0,1] based on the these results, which is partially generalized to three dimensional case. These curves represent well the numerical results of the conductivities.