Minimally Diluted Deterministic Rules

Substituting A/ = 9 into equation 7.92, we obtain the following iterative map for the density p  

For small enough values of p so that pf p) p for all 0 p 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/ 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p- 

Rule i 4, on the other hand, has both a linear and quadratic term, so that / (p = 0) 0 in general, and is therefore predicted to have a second order (or continuous) phase transition. Although the mean-field predictions are, of course, dimension-independent, they are expected to become exact as the dimension d —7 oo. In practice, it is often found that there exists a critical dimension dc above which the mean-field critical exponents are recovered exactly. 

An unstable fixed point, pu = pu p) appears forp pc and defines a boundary such that if p t = 0) pu p) the system decays to zero and if p t = 0) pu p) then p t — oo) = ps p), where ps p) is the stable fixed point for a PCA defined by probability p. Notice that since pu p) Pu p ) for p p, the value of the unstable fixed point for p = 1, Py = pu p = 1) 0.1577, represents a cutoff point for initial densities leading to a nonzero limit for any p i.e. if p t = 0) p[/ then p t — oo) = 0. Putting all of this together, we see that if we consider only those initial densities p t = 0) py then the poo versus p curve for rule i 2 exhibits a typical first-order phase transition at p = pc. The other rules are handled in the same fashion. 

---