Percolative conduction and the Alexander-Orbach conjecture in two dimensions

Alexander and Orbach have recently proposed that the ratio of the fractal dimensionality of the incipient infinite cluster in percolation to the fractual dimensionality of a random walk on the cluster is ⅔, independent of the spatial dimensionality of the system. As a consequence, they predict that the electrical conductivity exponent $\frac{t}{\nu }=0.9479\mathrm{}\dots$ in two dimensions, where $\nu$ is the correlation-length exponent. Our numerical data, which are obtained from large-lattice finite-size scaling calculations, give a value $\frac{t}{\nu }={0.973}_{-0.003\mathrm{}}^{+0.005\mathrm{}}$, in disagreement with the conjecture by 2.6%.