Subcritical monotone cellular automata

Abstract

We study monotone cellular automata (also known as $\mathcal{U}$-bootstrap percolation) in ${\mathbb{Z}}^d$ with random initial configurations. Confirming a conjecture of Balister, Bollobás, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non-zero for all subcritical models.

23 pages, 2 figures, added Theorem 7.1