Flow in porous media: The "backbone" fractal at the percolation threshold

We show that for all Euclidean dimensions $d \stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}={\overline{d}}_{w}\ensuremath{-}{\overline{d}}_{f}$, where ${L}_{R}\ensuremath{\sim}{\ensuremath{\xi}}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}}$ is the effective resistance between two points separated by a distance comparable with the correlation length $\ensuremath{\xi},{\overline{d}}_{f}$ is the fractal dimension of the backbone, and ${\overline{d}}_{w}$ is the fractal dimension of a random walk on the same backbone. We also find a relation between the backbone and the full percolation cluster, ${\overline{d}}_{w}\ensuremath{-}{\overline{d}}_{f}={d}_{w}\ensuremath{-}{d}_{f}$. Thus the Alexander-Orbach conjecture ($\frac{{d}_{f}}{{d}_{w}}=\frac{2}{3}$ for $dg~2$) fails numerically for the backbone.