The geometry of the percolation threshold

Computer‐generated pictures are presented of the connected component (’’infinite cluster’’) found at concentrations just above the threshold for 2D site percolation in large (400×400 site) lattices. For each case, we also show the ’’backbone’’ of the cluster, the smaller set of sites through which a current may flow. The simulations are contrasted with the model of conduction just above threshold due to Skal and Shklovskii and to de Gennes. That model is found to be inconsistent with the observed critical behavior of the conductivity in 2D and 3D models, but may apply to percolation in 4D and above. We show that a proper treatment of inhomogeneity on scales smaller than the coherence length is necessary to account for the observed conductivity and backbone volume just above threshold, and introduce a self‐similar model which accounts reasonably well for these properties.