Breakdown of Alexander-Orbach conjecture for percolation: Exact enumeration of random walks on percolation backbones

Abstract

We carry out the first exact enumeration studies of random walks on the percolation backbone. Using a relation between the backbone and the full cluster, we find for the $d=2$ conductivity exponent $\frac{t}{\ensuremath{\nu}}=0.970\ifmmode\pm\else\textpm\fi{}0.009$, which means that the Alexander-Orbach conjecture for percolation can hold only if our error bars were multiplied by a factor of 3. We also perform the first calculations of the chemical length exponent ${\overline{d}}_{l}$ that measures the dependence on $l$ of the number of backbone sites within a chemical distance $l$; we find ${\overline{d}}_{l}=1.44\ifmmode\pm\else\textpm\fi{}0.03$.