Geometric effects in continuous-media percolation

We study the simple system of a two-dimensional square lattice composed of good-conductor and poor-conductor squares, with the use of a clustered mean-field approximation. Instead of the well-known threshold behavior predicted by the two-component site percolation model or the effective-medium theory, we find two conductivity percolation thresholds at which the real and imaginary parts of the effective dielectric constant exhibit distinct critical behaviors. The cause of this double-threshold characteristic is shown to be the existence of a third conductivity scale arising from the corner-corner interactions between second-nearest-neighbor squares. Analogies with site percolation models are also detailed. It is demonstrated that as $\left|\frac{{\epsilon }_1}{{\epsilon }_2}\right|\to \infty$, where ${\epsilon }_{1\left(2\right)\mathrm{}}$ is the complex dielectric constant of the good (poor) conductor, the continuum system can be made equivalent to two versions of the square-lattice site percolation model, depending on whether $\left|{\epsilon }_1\right|\to \infty$ or $\left|{\epsilon }_2\right|\to 0$. The paper concludes with a discussion of possible implications for three-dimensional continuous-media percolating systems.