Universality for Monotone Cellular Automata

Abstract: 

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 attempt to motivate this conjecture by discussing some relatively simple special cases (known as bootstrap percolation), and general models in two dimensions (where very precise results are known). I will also discuss some potential applications to the Ising model of ferromagnetism, and kinetically constrained models of the liquid-glass transition. Finally, I will state a theorem which proves the conjecture, and moreover determines the value of r for every model.

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