Renormalization of cellular automata and self-similarity

TL;DR

This paper develops a renormalization framework for one-dimensional probabilistic cellular automata, revealing the instability of self-similarity in deterministic automata and identifying directed percolation as the key critical phenomenon.

Contribution

It introduces a general algebraic renormalization group method for cellular automata and applies it to analyze their critical behavior and self-similarity properties.

Findings

Renormalization fixed points for deterministic automata are unstable.

Large scale structure of self-similar deterministic automata is fragile under errors.

Directed percolation characterizes the non-trivial critical probabilistic automata.

Abstract

We study self-similarity in one-dimensional probabilistic cellular automata (PCA) using the renormalization technique. We introduce a general framework for algebraic construction of renormalization groups (RG) on cellular automata and apply it to exhaustively search the rule space for automata displaying dynamic criticality. Previous studies have shown that there exists several exactly renormalizable deterministic automata. We show that the RG fixed points for such self-similar CA are unstable in all directions under renormalization. This implies that the large scale structure of self-similar deterministic elementary cellular automata is destroyed by any finite error probability. As a second result we show that the only non-trivial critical PCA are the different versions of the well-studied phenomenon of directed percolation. We discuss how the second result supports a conjecture regarding the universality class for dynamic criticality defined by directed percolation.