The lower tail of triangle counts in sparse random graphs

Let $T_n$ be the number of triangles in the Erdős-Rényi random graph $G(n,p)$ with $n^{-1/2}\ll p\ll1$ and fix $η\in(0,1)$. We study the large deviation rate of the lower tail event $T_n\leη\mathbb{E}[T_n]$, and the typical structure of $G(n,p)$ conditioned on this event. By earlier work [16,19,21,4