**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 the first half of 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 relatively simple 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. In the second (more technical) half of the talk, I will give a rough overview of some of the new ingredients in the proofs.

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

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Based on

arxiv.org/abs/21https://arxiv.org/abs/2203.13806