Large deviations for subcritical bootstrap percolation on the random graph

We study atypical behavior in bootstrap percolation on the Erd\H{o}s-R\'enyi random graph. Initially a set $S$ is infected. Other vertices are infected once at least $r$ of their neighbors become infected. Janson et al. (2012) locates the critical size of $S$, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this note, we calculate the rate function for the event that a small set $S$ eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.