Percolation Theory of Nonlinear Circuit Elements

Percolation theory is extended to the case of circuit elements with nonlinear $I-V$ characteristics, particularly the special cases $V=I^{\alpha }\mathcal{r}$, which form the universality classes. Near the percolation threshold the "conductance" $\frac{I}{V^{\frac{1}{\alpha }}}$ vanishes like ${(p-p_c)}^t$, where $t$ depends on $\alpha$ and dimensionality. The Skal-Shklovskii-De Gennes model gives $t\left(\alpha \right)=(d-1\mathrm{})\mathrm{}\nu +\frac{(\varphi -\nu )}{\alpha }$, where $\varphi$ is only weakly dependent on $\alpha$ and $d$, and approaches unity at $d=6\mathrm{}$. Renormalization methods are used to study the $\alpha$ dependence of $t$ in two dimensions.