PCA — Spin Model Equivalence

Consider a size N one-dimensional lattice with site variables Si t) = 1, i = On even (or odd) time steps, either even-indexed (or odd-indexed) sites evolve according to fixed probabilistic peripheral rules that is, according to rules that depend only on the values of a given site s neighboring sites. Such rules are completely specified by a set of four conditional probabilities, 0 Ui 1, i = T...4  

Now consider an entire temporal history of this PCA. That is, consider the effective two-dimensional lattice that is formed by stacking successive one-dimensional layers on top of one another (see figure 7.4). Because of the Markovian nature of the evolution, the probability of this temporal history is given simply by 

Following Domany and Kinzel [domany84], we can simplify this product expansion considerably by rewriting P(5s 8182) in exponential form  

The sum is taken over each elementary plaquette in the lattice (see figure 

We thus have that the time evolution of the one-dimensional PCA system is equivalent to the equilibrium statistical mechanics of a spin model on a triangular lattice ([domany84], [geor89]).  

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