Subcritical bootstrap percolation via Toom contours

TL;DR

This paper offers a simpler proof that subcritical bootstrap percolation models have a positive critical probability in any dimension, using Toom contours, and improves bounds on Toom's automaton stability threshold.

Contribution

It introduces an alternative, more straightforward proof method based on Toom contours, enhancing bounds and extending results to all dimensions.

Findings

Proves positive critical probability for subcritical bootstrap percolation in any dimension.

Provides better bounds for Toom's North-East-Center majority rule automaton.

Simplifies the proof technique compared to previous multi-scale renormalisation methods.

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.