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

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In the previous article (S. Matsutani and Y. Shimosako and Y. Wang, Physica A 391 (2012) 5802-5809) we numerically investigated an electric potential problem with high contrast local conductivities (γ₀ and γ₁, 0<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 pc and gave an approximated curve representing a relation between the homogenized conductivity and the volume fraction p over [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→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] of the approximated curve to [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.