Renormalization group of the Domany-Kinzel cellular automaton

Tomé, Tânia; Oliveira, Mário J. de

doi:10.1103/physreve.55.4000

Cited by 14 publications

(7 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3 We note that while the percolation of the Domany-Kinzel automaton has been studied using a renormalization scheme similar to ours in e.g. [10], the larger generality in our scheme allows us to see explicitly that the automaton is only a fixed point if there exists a completely absorbing state, as shown both by the roots of the polynomials and by an analysis of the eigenvectors at the fixed point.…”

Section: Non-deterministic Fixed Pointsmentioning

confidence: 98%

Renormalization of Cellular Automata and Self-Similarity

Edlund

Jacobi

2010

J Stat Phys

View full text Add to dashboard Cite

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.

show abstract

Section: Non-deterministic Fixed Pointsmentioning

confidence: 98%

Renormalization of Cellular Automata and Self-Similarity

Edlund

Jacobi

2010

J Stat Phys

View full text Add to dashboard Cite

show abstract

“…Phenomenological renormalization group for the Domany-Kinzel model with mean-field of order 4 (adapted from Ref. [23]). The trajectories that show a circle at p = 0.5 finally end in the stable attracting point p = q = 0.…”

Section: J Phenomenological Renormalization Groupmentioning

confidence: 99%

Phase Transitions of Cellular Automata

Bagnoli

Rechtman

2018

Emergence, Complexity and Computation

View full text Add to dashboard Cite

We explore some aspects of phase transitions in cellular automata. We start recalling the standard formulation of statistical mechanics of discrete systems (Ising model), illustrating the Monte Carlo approach as Markov chains and stochastic processes. We then formulate the cellular automaton problem using simple models, and illustrate different types of possible phase transitions: density phase transitions of first and second order, damage spreading, dilution of deterministic rules, asynchronism-induced transitions, synchronization phenomena, chaotic phase transitions and the influence of the topology. We illustrate the improved mean-field techniques and the phenomenological renormalization group approach.

show abstract

“…(5). Previous work has shown the quality of the resulting renormalisation to be improved by incorporating the dynamics of the PCA, P, referred to as the dynamically driven renormalisation group (Vespignani et al 1996). In essence, the dynamics are utilised by taking a least-squares solution weighted by the stationary probability distribution of the 2k × 2k spins at the microscopic level, shown in Fig.…”

Section: Stationary Probability Distributionmentioning

confidence: 99%

“…1. This is a natural progression from the work of Edlund and Nilsson Jacobi (2010) who present a concise and systematic approach to renormalisation of such one-dimensional CA, which itself builds on previous work labelled as application of a dynamically driven renormalisation group (Toméand et al 1997;De Oliveira and Satulovsky 1997). It is known that in applying some scale transformation to a CA, we do not guarantee the existence of an exact coarse grained dynamics, one which exactly incorporates all possible microscopic transitions.…”

mentioning

confidence: 99%

Renormalisation of 2D Cellular Automata with an Absorbing State

Weaver

Prügel-Bennett

2015

J Stat Phys

View full text Add to dashboard Cite

We describe a real-space renormalisation scheme for non-equilibrium probabilistic cellular automata (PCA) models, and apply it to a two-dimensional binary PCA. An exact renormalisation scheme is rare, and therefore we provide a method for computing the stationary probability distribution of states for such models with which to weight the renormalisation, effectively minimising the error in the scale transformation. While a mean-field approximation is trivial, we use the principle of maximum entropy to incorporate nearest-neighbour spin-correlations in the steady-state probability distribution. In doing so we find the fixed point of the renormalisation is modified by the steady-state approximation order.

show abstract

Renormalization group of the Domany-Kinzel cellular automaton

Cited by 14 publications

References 15 publications

Renormalization of Cellular Automata and Self-Similarity

Renormalization of Cellular Automata and Self-Similarity

Phase Transitions of Cellular Automata

Renormalisation of 2D Cellular Automata with an Absorbing State

Contact Info

Product

Resources

About