An Investigation of the Phase Transitions of a Family of Probabilistic Automata

We investigate a family of totalistic probabilistic cellular automata (PCA) which depend on three parameters. For the uniform random neighborhood and for the symmetric 1D PCA the exact stationary distribution is computed for all finite n. This result is used to evaluate approximations (uni-variate and bi-variate marginals). It is proven that the uni-variate approximation (also called mean-field) is exact for the uniform random neighborhood PCA. The exact results and the approximations are used to investigate phase transitions. We compare the results of two order parameters, the uni-variate marginal and the normalized entropy. Sometimes different transitions are indicated by the Ehrenfest classification scheme. This result shows the limitations of using just one or two order parameters for detecting and classifying major transitions of the stationary distribution. Furthermore, finite size scaling is investigated. We show that extrapolations to n=∞ from numerical calculations of finite n can be misleading in difficult parameter regions. Here, exact analytical estimates are necessary.