Theory and Applications of Random Cellular Automata and Related Models

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9971543Griffeath On the theoretical side, the present proposal will further develop the investigator's rigorous theory of threshold growth, and related dynamics, for deterministic and probabilistic cellular automata (CA). The theory of growth models has played a key role in the analysis of spatial complex systems, dating back to the middle of the century. During the present funding period, Griffeath will seek to further that theory by focusing on five main research projects: scaling laws for critical monotone solidification, shape theory for random perturbations of threshold growth, a prototype of spinodal clustering, quantitative analysis of 'small-world networks', and asymptotics for von Koch - type CA solidification. These studies are primarily theoretical, their goal being rigorous mathematical results. On the experimental side, Griffeath has begun to examine more complex interactions which exhibit a wide variety of intriguing self-organizational effects not present in the simplest "linear" spatial systems. Preliminary empirical modeling projects at present include digital biofilms, nonlinear spatial modeling of social interaction, CA pattern dynamics, and a simple CA paradigm for traffic jams. In broad strokes, the ongoing research of David Griffeath combines mathematical analysis and computer visualization in the study of complex spatial systems. The investigator exploits this interplay for the theoretical and empirical study of deterministic and random cellular automata, local lattice dynamics which serve as prototypes for spiral formation in excitable media, the morphology of crystals, and various other nonlinear processes such as nucleation, flocking, host-parasite interactions, dendritic growth, and phase separation by surface tension. The analysis of aggregation, shape, interfaces, and droplet interaction for such basic growth models sheds new light on many areas of applied science. The investigator's research contributes to the mathematical foundations of such processes.