A percolational criterion for the Anderson transition in structurally disordered systems

The condition for the Anderson transition in systems where the atoms are completely randomly distributed in space with concentration c is obtained by assuming the coincidence of the onset of extended states and of the infinite cluster in a special percolation model. This model consists of the random system of sites with the probability of connection between sites depending on the distance between them and on their local surroundings. The thresholds of percolation in this model, determined in the three-dimensional case by considering the dependence of the average numbers of sites connected to a given site on the configuration of unbroken connections and by direct computer simulation are in good agreement. They yield for a parameter M=c -1/3 alpha (where alpha -1 is a characteristic length of the exponentially decreasing transfer integral) the critical value M c =5.8+or-0.4. In two dimensions the model predicts a transition for M c =3.9+or-0.2.