Crossover behavior of conductivity in a discontinuous percolation model

When conducting bonds are occupied randomly in a two-dimensional square lattice, the conductivity of the system increases continuously as the density of those conducting bonds exceeds the percolation threshold. Such a behavior is well known in percolation theory; however, the conductivity behavior has not been studied yet when the percolation transition is discontinuous. Here we investigate the conductivity behavior through a discontinuous percolation model evolving under a suppressive external bias. Using effective medium theory, we analytically calculate the conductivity behavior as a function of the density of conducting bonds. The conductivity function exhibits a crossover behavior from a drastically to a smoothly increasing function beyond the percolation threshold in the thermodynamic limit. The analytic expression fits well our simulation data.

Figure 1

(a) Schematic diagram of circuit structure for $p\ge p_{cm}$ for the SCA model that consists of bonds of unit resistance. The occupied bonds are classified into original bridge bonds (thick red resistors) and original nonbridge bonds (thin blue resistors). (b) We simplify the whole circuit as series connection of a bundle of original bridge bonds of unit resistance and two compact clusters consiting of bonds of resistance $r_a$ by applying effective medium theory. (c) The combined resistance of two compact clusters is calculated as $r_a^{\prime }=r_a(L-1)/L\approx r_a$ for large $L$, and the combined resistance of original bridge bonds $r_b$ is calculated as $r_b=1/L{p}_b$, where the derivations are shown in the main text. We can calculate the conductivity as $g_m\left(p\right)=1/(r_a^{\prime }+r_b)$.

Figure 2

Plot of $G_m\left(p\right)$ and $g_m\left(p\right)$ vs $p$. $G_m\left(p\right)$ is the fraction of nodes belonging to the spanning cluster. $G_m\left(p\right)$ jumps to $G_1\left(p\right)$ from zero at $p_{cm}$ and follows the envelope of $G_1\left(p\right)$ after that. $g_m\left(p\right)$ is the conductivity; it becomes positive at $p_{cm}$ and grows drastically after that. As $m$ is increased, $p_{cm}$ is delayed. Data are shown for $m=1$ (red), $m=2$ (yellow), $m=3$ (green), $m=4$ (blue), and $m=5$ (purple) from left to right. $L=300$ are considered. Results are for a single sample.

Figure 3

Plot of $g_m\left(p\right)$ vs $p$ for one sample with $L=300$. Just after $p_{cm}$, $g_m\left(p\right)$ becomes positive and grows drastically. Red solid lines are obtained from Eq. (3) for $m=2$ (a), $m=3$ (b), $m=4$ (c), and $m=5$ (d). We can find that the theoretical formula fits the simulation data well when $m$ is larger than the tricritical point $m_c\approx 2.55$.