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 $d=2\mathrm{}$ conductivity exponent $\frac{t}{\nu }=0.970\mathrm{}\pm 0.009$, 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 ${\overline{d}}_l=1.44\mathrm{}\pm 0.03$.