Renormalization of Cellular Automata and Self-Similarity
Journal article, 2010

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.



Cellular automata


Directed percolation



Erik Edlund

SuMo Biomaterials

Chalmers, Energy and Environment, Physical Resource Theory

Martin Nilsson Jacobi

Chalmers, Energy and Environment, Physical Resource Theory

Journal of Statistical Physics

0022-4715 (ISSN) 1572-9613 (eISSN)

Vol. 139 6 972-984

Subject Categories

Physical Sciences



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