Abstract: See http://www.math.ubc.ca/~holroyd/boot/ for a picture.

Cellular automata arise naturally in the study of physical systems, and exhibit a seemingly limitless range of intriguing behaviour. Such models lend themselves naturally to computer simulation, but rigorous analysis can be notoriously difficult, and can yield highly unexpected results.

Bootstrap percolation is a very simple model for nucleation and growth which turns out to hold many surprises. Sites in a square grid are initially declared "infected" independently with some fixed probability. Subsequently, healthy sites become infected if they have at least two infected neighbours, while infected sites remain infected forever. The model undergoes a phase transition at a certain threshold whose asymptotic value differs from numerical predictions by more than a factor of two! This discrepancy points to a previously unsuspected phenomenon called "crossover", and leads to further intriguing questions.