Discrete Excitable Media and Finding a Place to Park
David Sivakoff, Department of Statistics and Department of Mathematics, The Ohio State University
Two classical discrete, deterministic models of excitable media are the cyclic cellular automaton (CCA) and Greenberg-Hastings models, which model rock-paper-scissors-like competition of species and neural networks, respectively. Both models have been extensively studied on the cubic lattice, Z^d, where one is typically interested in whether or not sites are excited (change states) infinitely often, and if so, whether the states of nearby sites tend toward agreement. We introduce a new comparison process for the 3-species variants of these models, which allows us to study the asymptotic rate at which a site gets excited. Using this comparison process, we also analyze a new model for pulse-coupled oscillators in one dimension, introduced recently by Lyu, called the firefly cellular automaton (FCA). When the number of species exceeds 3, the CCA dynamics become significantly more complex, and we turn to a simplified model called the "parking process," for which we can prove existence of a sharp phase transition. This talk will be based on joint works with Damron, Gravner, Junge and Lyu.
Note: Seminars are free and open to the public. Reception to follow.