Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions

Abstract

We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r \geq 2$: that is, an initial set of density $p$ almost surely percolates $\mathbb{Z}^d$ for every $p>0$. These results effectively ended the study of bootstrap percolation on infinite lattices. Recently Bollob\'as, Smith and Uzzell introduced a broad class of percolation models called $\mathcal{U}$-bootstrap percolation, which includes $r$-neighbour bootstrap percolation as a special case. They divided two-dimensional $\mathcal{U}$-bootstrap percolation models into three classes -- subcritical, critical and supercritical -- and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical $\mathcal{U}$-bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^2$. They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical $\mathcal{U}$-bootstrap percolation model has a non-trivial critical probability on $\mathbb{Z}^2$. This is new except for a certain `degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.