Classification of backward filtrations and factor filtrations: examples from cellular automata

We consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik’s theory, but we also study them from another point of view that takes into account the measure-preserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.

Parallel Simulation of Two-Dimensional Ising Models Using Probabilistic Cellular Automata

We study the numerical simulation of the shaken dynamics, a parallel Markovian dynamics for spin systems with local interaction and transition probabilities depending on the two parameters q and J that “tune” the geometry of the underlying lattice. The analysis of the mixing time of the Markov chain and the evaluation of the spin-spin correlations as functions of q and J, make it possible to determine in the (q, J) plane a phase transition curve separating the disordered phase from the ordered one. The relation between the equilibrium measure of the shaken dynamics and the Gibbs measure for the Ising model is also investigated. Finally two different coding approaches are considered for the implementation of the dynamics: a multicore CPU approach, coded in Julia, and a GPU approach coded with CUDA.

Bistability and time crystals in long-ranged directed percolation

Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections. Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata.

A cellular automata rule placing a maximal number of dominoes in the square and diamond

The objective is to demonstrate that a probabilistic cellular automata rule can place reliably a maximal number of dominoes in different active area shapes, exemplarily evaluated for the square and diamond. The basic rule forms domino patterns, but the number of dominoes is not necessarily maximal and the patterns are not always stable. It works with templates derived from domino tiles. The first proposed enhancement (Rule Option 1) can form always stable patterns. The second enhancement (Rule Option 2) can maximize the number of dominoes, but the reached patterns are not always stable. All rules drive the evolution by specific noise injection.

About Cellular Automata

The chapter describes the basic theoretical principles for the theory of cellular automata. The history of the emergence of cellular automata based on an analysis of existing information sources is presented. The modern classification of cellular automata is presented. The structures of elementary and two-dimensional cellular automata are described. In terms of the rules for the functioning of cellular automata, synchronous, asynchronous, and probabilistic cellular automata are briefly described. Researchers are presented who have made a significant contribution to the development of the theory of cellular automata.