Asynchronous cellular automata

This text has been proposed for the Encyclopedia of Complexity and Systems Science edited by Springer Nature and should appear in 2018.International audienceThis text is intended as an introduction to the topic of asynchronous cellular automata. We start from the simple example of the Game of Life and examine what happens to this model when it is made asynchronous (Sec. 1). We then formulate our definitions and objectives to give a mathematical description of our topic (Sec. 2). Our journey starts with the examination of the shift rule with fully asynchronous updating and from this simple example, we will progressively explore more and more rules and gain insights on the behaviour of the simplest rules (Sec. 3). As we will meet some obstacles in having a full analytical description of the asynchronous behaviour of these rules, we will turn our attention to the descriptions offered by statistical physics, and more specifically to the phase transition phenomena that occur in a wide range of rules (Sec. 4). To finish this journey, we will discuss the various problems linked to the question of asynchrony (Sec. 5) and present some openings for the readers who wish to go further (Sec. 6)

Local structure approximation as a predictor of second order phase transitions in asynchronous cellular automata

International audienceThe mathematical analysis of the second-order phase transitions that occur in α-asynchronous cellular automata field is a highly challenging task. From the experimental side, these phenomena appear as a qualitative change of behaviour which separates a behaviour with an active phase, where the system evolves in a stationary state with fluctuations, from a passive state, where the system is absorbed in a homogeneous fixed state. The transition between the two phases is abrupt: we ask how to analyse this change and how to predict the critical value of the synchrony rate α. We show that an extension of the mean-field approximation, called the local structure theory, can be used to predict the existence of second-order phase transitions belonging to the directed percolation university class. The change of behaviour is related to the existence of a transcritical bifurcation in the local structure maps. We show that for a proper setting of the approximation, the form of the transition is predicted correctly and, more importantly, an increase in the level of local structure approximation allows one to gain precision on the value of the critical synchrony rate which separates the two phases