Subcritical $\mathcal {U}$-bootstrap percolation models have non-trivial phase transitions
Subcritical $\mathcal {U}$-bootstrap percolation models have non-trivial phase transitions
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 (1987) (in the case $d=r=2$) and Schonmann (1992) (for all $d \geqslant r \geqslant 2$) …