Percolation theory for nonlinear conductors

Under broad conditions, a network of nonlinear conductors has an $I-V$ characteristic uniquely determined by Kirchhoff's rules. By means of a renormalization calculation, we show that near the percolation threshold the details of the microscopic $I-V$ characteristic are averaged out, so that the bulk material approaches power-law conductor behavior ($V=I^{\alpha }$). The threshold exponents $t\left(\alpha \right)$ and $s\left(\alpha \right)$ are discussed in the limiting cases of two dimensions (where they are related by duality) and high dimensionality (by solving the Cayley-tree model).