Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees

In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every stationary measure $\nu$ of the PCA we can associate a space-time Gibbs measure $\mu_{\nu}$ on $\mathbb{Z} \times \mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true. A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCA on d-ary trees for $d\in \{ 1,2,3\}$ and characterizing the invariant product Bernoulli measures.Comment: 17 page

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let \b denote the inverse temperature and let \L_\b \subset \Z^2 be a square box with periodic boundary conditions such that \lim_{\b\to\infty}|\L_\b|=\infty. We run the dynamics on \L_\b starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large \b, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in \L_\b. Using potential-theoretic methods, we compute the \emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as \b\to\infty. It turns out that this time grows as Ke^{\Gamma\b}/|\L_\b| for Glauber dynamics and K\b e^{\Gamma\b}/|\L_\b| for Kawasaki dynamics, where $\Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |\L_\b|). The fact that the average nucleation time is inversely proportional to |\L_\b| is referred to as \emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition \L_\b.Comment: 45 pages, 11 figure

Competitive nucleation in metastable systems

Metastability is observed when a physical system is close to a first order phase transition. In this paper the metastable behavior of a two state reversible probabilistic cellular automaton with self-interaction is discussed. Depending on the self-interaction, competing metastable states arise and a behavior very similar to that of the three state Blume-Capel spin model is found

Basic Ideas to Approach Metastability in Probabilistic Cellular Automata

Cellular Automata are discrete--time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata, are discrete time Markov chains on lattice with finite single--cell states whose distinguishing feature is the \textit{parallel} character of the updating rule. We review some of the results obtained about the metastable behavior of Probabilistic Cellular Automata and we try to point out difficulties and peculiarities with respect to standard Statistical Mechanics Lattice models.Comment: arXiv admin note: text overlap with arXiv:1307.823

Nucleation for one-dimensional long-range Ising models

In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration -1 is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet can be macroscopic or mesoscopic, according to the value of the external magnetic field.Comment: 15 pages, 3 figure

Homogeneous and heterogeneous nucleation in the three--state Blume--Capel model

The metastable behavior of the stochastic Blume--Capel model with Glauber dynamics is studied when zero-boundary conditions are considered. The presence of zero-boundary conditions changes drastically the metastability scenarios of the model: \emph{heterogeneous nucleation} will be proven in the region of the parameter space where the chemical potential is larger than the external magnetic field.Comment: 25 pages, 18 figure

Phase transitions in random mixtures of elementary cellular automata

We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models

Competitive nucleation in reversible Probabilistic Cellular Automata

The problem of competitive nucleation in the framework of Probabilistic Cellular Automata is studied from the dynamical point of view. The dependence of the metastability scenario on the self--interaction is discussed. An intermediate metastable phase, made of two flip--flopping chessboard configurations, shows up depending on the ratio between the magnetic field and the self--interaction. A behavior similar to the one of the stochastic Blume--Capel model with Glauber dynamics is found