Moderate deviations of triangle counts – the lower tail

Abstract

Two recent papers~\cite{GGS} and~\cite{NRS22} study the lower tail of triangle count deviations in random graphs $G(n,m)$ with positive density $t:=m/\binom{n}{2}\in (0,1)$. Let us write $D_{\triangle}(G)$ for the deviation of the triangle count from its mean. Results of~\cite{GGS} and~\cite{NRS22} determine the order of magnitude of the log probability $\log(\pr{D_{\triangle}(G(n,m)) \, < \, - \tau\binom{n}{3}})$ for the ranges $n^{-3/2}\ll \tau\ll n^{-1}$ and $n^{-3/4}\ll\tau\ll 1$ respectively. Furthermore, in~\cite{NRS22} it is proved that the log probability is at least $\Omega(\tau^2 n^{3})$ in the ``missing'' range $n^{-1}\ll \tau\ll n^{-3/4}$, and they conjectured that this result gives the correct order of magnitude. Our main contribution is to prove this conjecture.