Subcritical bootstrap percolation via Toom contours

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 [18] of the classical framework of Toom [20]. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions [1, 2], 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.