Northwest Probability Seminar 2014 – Sharp higher order corrections for the critical value of a bootstrap percolation model

Bootstrap Percolation (BP) models are simple cellular automata with a deterministic growth rule and random initial configuration. It is known that (critical) BP models on the lattice undergo a metastable phase transition on a lattice of fixed size, as the density of the initial configuration increases. Holroyd (2004) determined sharp first order asymptotics for the critical value for the canonical nearest-neighbor model in 2D. Surprisingly, this value turned out to be very far removed from computer simulations of BP on large lattices, suggesting that higher order corrections to the critical value may dominate even on large lattices. We study the scaling of the critical value for the so-called anisotropic (1,2)-model. Duminil-Copin and van Enter recently showed sharp first order asymptotics for the critical value of this model. We determine sharp second and third order corrections, and show that they are both large enough to dominate the critical value of anisotropic bootstrap percolation on any scale that is feasibly accessed by computers. Based on joint work with Hugo Duminil-Copin, Aernout van Enter, and Robert Morris.