Series analysis of randomly diluted nonlinear networks with negative nonlinearity exponent

The behavior of randomly diluted networks of nonlinear resistors, for each of which the voltage-current relationship is $\mathrm{}\left|V\right|=r{\mathrm{}\left|I\right|\mathrm{}}^{\alpha }$, where $\alpha$ is negative, is studied using low-concentration series expansions on $d$-dimensional hypercubic lattices. The average nonlinear resistance $〈R〉$ between a pair of points on the same cluster, a distance $r$ apart, scales as $r^{\frac{\zeta \left(\alpha \right)}{\nu }}$, where $\nu$ is the correlation-length exponent for percolation, and we have estimated $\zeta \left(\alpha \right)$ in the range $-1\mathrm{}\le \alpha \le 0$ for $1\le d\le 6$. $\zeta \left(\alpha \right)$ is discontinuous at $\alpha =0\mathrm{}$ but, for $\alpha <0\mathrm{}$, $\zeta \left(\alpha \right)$ is shown to vary continuously from ${\zeta }_{max}$, which describes the scaling of the maximal self-avoiding-walk length (for $\alpha \to 0-$), to ${\zeta }_{\mathrm{BB}}$, which describes the scaling of the backbone (at $\alpha =-1\mathrm{}$). As $\alpha$ becomes large and negative, the loops play a more important role, and our series results are less conclusive.