MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Critical Probabilities and Convergence Time of Stavskaya's Probabilistic Cellular Automata

Lorenzo Taggi


We consider a class of probabilistic cellular automata undergoing a phase transition with an absorbing-state. Denoting by ${\mathcal{U}} (s)$ the neighbourhood of the site $s$, the transition probability is $T(\eta_s = 1 | \eta_{{\mathcal{U}}(s)}) = 0$ if $\eta_{{\mathcal{U}}(s)}= \mathbf{0}$ or $p$ otherwise, $\forall s \in \mathbb{Z}$. For any $\mathcal{U}$ there exists a non-trivial critical probability $p_c( {\mathcal{U}})$ which separates a phase with an absorbing-state from a fluctuating phase. We study how the neighbourhood affects the value of $p_c( {\mathcal{U}})$ and we provide lower bounds for $p_c( {\mathcal{U}})$. Furthermore, using techniques of dynamic renormalization, 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 appears as an open problem in Toom et al. (2004, 1995, 1990).

MSC Codes:
82C27, 82C22, 82C20

Related publications

2015 Repository Open Access
Lorenzo Taggi

Critical probabilities and convergence time of percolation probabilistic cellular automata

In: Journal of statistical physics, 159 (2015) 4, pp. 853-892