Jan Swart

Peierls bounds from Toom contours

In 1980, Andrei Toom proved his celebrated stability theorem, which says that the upper invariant law of a cellular automaton on the d-dimensional integer lattice is stable under small random perturbations if and only if the automaton is an eroder, which means that the unperturbed system started with finitely many zeros returns to the all-one state after a finite number of steps. Import open problems remain for random cellular automata and their continuous-time counterparts, monotone interacting particle systems. For example, while Toom proved stability of the discrete-time North-East-Center majority voting rule, also known as Toom's rule, stability of the interacting particle system with Toom's rule is an unresolved problem. At the heart of Toom's proof lies an intricate Peierls argument. I will explain a reformulated and extended version of this argument that can be used to derive lower bounds on the density of the upper invariant law for a variety of systems, including systems with intrinsic randomness and some interacting particle system. This is joint work in progress with Réka Szabó and Cristina Toninelli.