E - Statistical Weights in the DDRG

Summary

There are four different sets of configurations (see Figure E. 1). The weight factor W_α can be written as a product of the probability that _α sites are occupied (and 4 - α are unoccupied) by a degeneracy factor n_α: W_α = n_αρ^α (1 - ρ)^4-α. The expressions for n_α are given in Figure E.1.

The calculations are performed as follows. Let us first consider the α = 2 case. There are K_4,2 = 6 configurations for which two of the four sites are critical. Two of these configurations correspond to the critical sites being placed along the diagonal. Such configurations are ignored because they do not meet the spanning criterion. This leaves us with four configurations. For each of these remaining configurations, we have two possible choices when we turn one of the critical sites into an unstable site. In total there are eight different but dynamically equivalent configurations of α = 2. Next, we turn our attention to α = 3 configurations. We can pick three critical sites in K_4,3 = 4 different

ways. The α = 3 sites break up into two classes of different dynamical behavior depending on where we choose to position the unstable site among the three critical sites. There are four configurations with the unstable site positioned between the two critical sites (see Figure E.1) and eight configurations where the critical site is located at either end of the string of three sites.