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

We show that for all Euclidean dimensions $d \tilde{\zeta }={\overline{d}}_w-{\overline{d}}_f$, where $L_R\sim {\xi }^{\tilde{\zeta }}$ is the effective resistance between two points separated by a distance comparable with the correlation length $\xi ,{\overline{d}}_f$ 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, ${\overline{d}}_w-{\overline{d}}_f=d_w-d_f$. Thus the Alexander-Orbach conjecture ($\frac{d_f}{d_w}=\frac{2}{3}$ for $d>~2$) fails numerically for the backbone.