Subcritical \(\mathcal{U}\)-bootstrap percolation models have non-trivial phase transitions (Q2796096)

The classical \(r\)-neighborhood bootstrap percolation model is formulated as follows. Given a graph \(G=(V,E)\), a subset \(A\subset V\) of the set of vertices of \(G\) is chosen by including vertices at random and independently with probability \(p\). The vertices in \(A\) are called infected and they remain infected forever, while uninfected vertices become infected if at least \(r\) of their neighbors are infected. Let \(A_0= A\) and let NEWLINE\[NEWLINEA_{t+1}=A_t\cup \{v\in V: | N(v)\cap A_t|\geq r\}, \;\;t=0,1,2,\dots.NEWLINE\]NEWLINE Thus, \(\lfloor A\rfloor:=\bigcup_{t=0}^\infty A_t\) is the set of vertices that become eventually infected. If \(\lfloor A\rfloor=V,\) it is said that \(A\) percolates, which means that bootstrap percolation assumes that \(A\) percolates if it infects all vertices of \(G\), but not just an infinite cluster, as in the case of classical percolation. Extending the aforementioned model, \textit{B. Bollobás} et al. [``Monotone cellular automata in a random environment'', Combinatorics, Probab. Comput. 24, No. 4, 687--722 (2015; \url{doi:10.1017/s0963548315000012})] introduced \({\mathcal U}\)-bootstrap percolation on \(V={\mathcal Z}^2\) in which new infections are made according to any rule that is local (= depends on finite neighborhoods), homogeneous and monotone, and they divided the models into three classes which are supercritical, critical and subcritical. The present paper proves that in the case of subcritical \({\mathcal U}\) and \(A\sim \mathrm{Bin}({\mathcal Z}^2,p)\), NEWLINE\[NEWLINE{\mathcal P}_p(0\in \lfloor A\rfloor )\to 0\text{ as } p\to 0NEWLINE\]NEWLINE and \(p_c({\mathcal Z}^2,{\mathcal U})>0,\) where \(p_c\) denotes the critical probability. This result was conjectured in the aforementioned paper.