Subcritical bootstrap percolation via Toom contours

Abstract

In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom's North-East-Center majority rule cellular automaton.

Subcritical monotone cellular automata

“…, and observe that h i (x ′ ) ≠ h i+1 (x ′ ), where x ′ is the projection of x onto {u} ⊥ , since x ∈ Q 1 ⊂ P, and we assumed that P does not intersect Σ + 4𝛾 ⋅ L (i) u . It follows, by (14), that there exists z ∈ Z i with 𝑑(x ′ , z) < 2 −6 𝜋 ⋅ g i . Let y ∈ Y be such that z is the orthogonal projection of y onto {u} ⊥ , and let Q 2 be the (i)-cube containing y.…”

“…The main aim of this paper is to prove the following theorem, which confirms one direction of this conjecture. We remark that an alternative (very different) proof of this theorem has been given by Hartarsky and Szabó [14], using a method that was developed recently by Swart, Szabó and Toninelli [16].…”