Bootstrap Percolation and Cellular Automata

We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics. We characterize the monotone automata that almost surely fill the space starting from any nontrivial Bernoulli measure. In contrast, we show the problem is undecidable if the monotonicity condition is dropped. We also construct examples where the space-filling property depends on the initial Bernoulli measure in a non-monotone way.