Critical probabilities and convergence time of Percolation Probabilistic Cellular Automata

Abstract

This paper considers a class of probabilistic cellular automata undergoing a phase transition with an absorbing state. Denoting by $\mathcal{U}(x)$ the neighbourhood of site $x$, the transition probability is $T({\eta }_x=1|{\eta }_{\mathcal{U}(x)})=0$ if ${\eta }_{\mathcal{U}(x)}=0$ or $p$ otherwise, $\forall x\in \mathbb{Z}$. For any $\mathcal{U}$ there exists a non-trivial critical probability $p_c(\mathcal{U})$ that separates a phase with an absorbing state from a fluctuating phase. This paper studies how the neighbourhood affects the value of $p_c(\mathcal{U})$ and provides lower bounds for $p_c(\mathcal{U})$. Furthermore, by using dynamic renormalization techniques, we prove that the expected convergence time of the processes on a finite space with periodic boundaries grows exponentially (resp. logarithmically) with the system size if $p>p_c$ (resp. $p<p_c$). This provides a partial answer to an open problem in Toom et al. (1990, 1994).

arXiv comment50 pages, 19 Figures

References in corpus (8)

• Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States
• Metastability for reversible probabilistic cellular automata with self--interaction
• Universal scaling behavior of directed percolation and the pair contact process in an external field
• Critical droplets in Metastable States of Probabilistic Cellular Automata
• A Monte Carlo investigation of the critical behavior of Stavskaya's probabilistic cellular automaton
• Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions
• Low-density series expansions for directed percolation III. Some two-dimensional lattices
• Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

Cited by in corpus (3)

• Local structure approximation as a predictor of second order phase transitions in asynchronous cellular automata
• Phase transitions in random mixtures of elementary cellular automata
• Generalised oriented site percolation