Moderate deviations of triangle counts in sparse Erd\H{o}s-R\'enyi random graphs $G(n,m)$ and $G(n,p)$

We consider the question of determining the probability of triangle count deviations in the Erd\H{o}s-R\'enyi random graphs $G(n,m)$ and $G(n,p)$ with densities larger than $n^{-1/2}(\log{n})^{1/2}$. In particular, we determine the log probability $\log\mathbb{P}(N_{\triangle}(G)\, >\, (1+\delta)p^3n^3)$ up to a constant factor across essentially the entire range of possible deviations, in both the $G(n,m)$ and $G(n,p)$ model. For the $G(n,p)$ model we also prove a stronger result, up to a $(1+o(1))$ factor, in the non-localised regime. We also obtain some results for the lower tail and for counts of cherries (paths of length $2$).