Screening of Deeply Invaginated Clusters and the Critical Behavior of the Random Superconducting Network

Abstract

Starting with an expression for the fractal dimension $d_u$ of the unscreened perimeter of an arbitrary fractal of dimension $d_f$, there are derived for the random superconducting network the results $\tilde{s}=(2-d)+d_u$, from which follow ${\tilde{\varphi }}_s=d_u$ and $d_w=d-d_u$. Here $\tilde{s}$ is the conductivity exponent, ${\tilde{\varphi }}_s$ the conductance exponent, and $d_w$ the fractal dimension of a random walk on the network. For $d=2\mathrm{}$, these results differ from the Alexander-Orbach conjecture by 0.3%.