A study of Probabilistic Cellular Automata

Abstract

In this thesis, we aim to review the existing literature to understand theoretical foundations of Probabilistic Cellular Automaton(PCAs) and draw out a condensed theory on the ergodicity behavior of the PCAs. On one hand we have surveyed this theory with examples and their connections in various other areas of mathematics. In the first part our study focuses on the intractability of computing a unique invariant measure for an ergodic PCA. We show via two calculations in chapter 4 that this is complex even for the case when we have a simple Bernoulli measure as an invariant measure of the PCA. On the other hand, we have discussed a method that have recently taken the center stage in proving the ergodicity of a certain class of PCAs known as the hard-core PCAs. This method known as method of Random walk has been a relatively easier way of showing the ergodicity. We have extended a result from paper "Ergodicity of the hard-core PCA with a random walk method" in specific parameter regime to show the application of the above method. In conclusion we have mentioned some future directions to unify several seemingly unrelated concepts that may give some new insight in proving the ergodicity of various other PCAs.