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

Starting with an expression for the fractal dimension dᵤ of the unscreened perimeter of an arbitrary fractal of dimension df, there are derived for the random superconducting network the results ̃ \~s=(2-d)+dᵤ, from which follow ̃ \~φₛ=dᵤ and dw=d-dᵤ. Here ̃ \~s is the conductivity exponent, ̃ \~φₛ the conductance exponent, and dw the fractal dimension of a random walk on the network. For $d=2$, these results differ from the Alexander-Orbach conjecture by 0.3%.