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

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 t =0.970 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 d l that measures the dependence on l of the number of backbone sites within a chemical distance l; we find d l=1.44 0.03.