A large deviation approach to super-critical bootstrap percolation on the random graph $G_{n,p}$

We consider the Erd"{o}s–R'{e}nyi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^{\ast }$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n-A_n^{\ast }}{f(n)}{\}}_{n\ge 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible range.