Among the most convenient models of physical processes are deterministic and random cellular automata. These mathematical objects describe configurations on lattices which evolve by a repeated update of a local rule. They have been utilized to model natural phenomena and have offered insights into fundamental organizational principles in many scientific fields, including physics, biology, computer science, and social sciences. From a more abstract perspective, cellular automata are a suitable tool used to characterize and catalog local dynamics which generate a prescribed global phenomenon. The first aim of this project is to develop and use techniques from modern probability theory, as well as computational tools, to study diverse aspects of cellular automata. Then second aim is to use the gathered insights to investigate more complex dynamics of diffusive, nonlocal and high-dimensional models inspired by crystal growth, genetics and social networks.
Three themes connect cellular automata with probability theory. The first issue, motivated by studying self-organization, involves evolution of a given rule from random initial states. The addressed topics include: nucleation, that is, random formation of centers that orchestrate a takeover of the available space; highly dependent percolation structures that determine the behavior of typical trajectories; and Lyapunov stability analysis through large deviations of branching random walks. The second direction are random rules, with a branching process approach to formation of stable periodic structures. The third, classical, theme is perturbation of deterministic updates by random noise. For more general models, the project aims to study non-local dynamics, focusing on nucleation theory, the nature of phase transitions, effects of space heterogeneity, and the boundary between short- and long-range interactions. Computation is a significant component of the research and involves simulations, analysis of algorithms, computer-aided proofs, numerical and statistical methods, and visualization techniques.
