Universality for Monotone Cellular Automata

The Universality Conjecture of Bollobás, Duminil-Copin, Morris and Smith states that every d-dimensional monotone cellular automaton is a member of one of d+1 universality classes, which are characterized by their behaviour on sparse random sets. More precisely, it states that if sites are initially infected independently with probability p, then the expected infection time of the origin is either infinite, or is a tower of height r for some r \in {1,...,d}.

In this talk I will state a theorem which proves the conjecture, and moreover determines the value of r for every model. I will also attempt to motivate this theorem by discussing some interesting (and well-studied) special cases, and some potential applications to non-monotone models such as the Ising model of ferromagnetism, and kinetically constrained models of the liquid-glass transition.

Joint work with Paul Balister, Béla Bollobás and Paul Smith.