Professor Gravner will pursue further research in cellular automata. The research will continue in several new directions. The growth models featured in this project range from simple threshold rules and competition dynamics, to interacting systems of random walks and percolation models in low- and high-dimensional spaces. Among the principle topics are: effects of random perturbations in the environment and in the rule, fluctuations of random interfaces, generation of fractal patterns, behavior of large--neighborhood systems, large scale limits of systems of random walks, and relationship between structure of genotype space and genetic diversity. The aim of this project is to understand microscopic and macroscopic principles by which various physical systems propagate disturbances far from the equilibrium. Among instances of such growth dynamics are crystal growth, spread of epidemics, propagation of waves, competition between species, and genetic diversification. Such research can contribute to understanding of such natural phenomena as: threshold density for survival and growth vs. the range of interaction for species in an unfriendly environment, transition between sharp geometric crystal shapes and rough random ones, effects of noisy environments on spread of infections, validity of approximation based on large range of interaction or fast diffusion, and role of neutral mutations in genetic drift.
