Sumsets of random sets

Given $m\in \mathbb{N}$ and a $p$-random subset $A\subseteq \mathbb{N}$, we asymptotically determine $\log \mathrm{Pr}(\mathrm{\mid }\mathbb{N}\setminus (A+A)\mathrm{\mid }\ge m)$ for $p$ above the threshold for this property. The proof is based on a bespoke container argument.