Moderate deviations of triangle counts in the Erdős-Rényi random graph G(n,m): The lower tail

Abstract

Let $N_{△}\left(G\right)$ be the number of triangles in a graph $G$. In Goldschmidt et al. (2020) and Neeman et al. (2023) (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs $G_{\left(m\right)}\sim G(n,m)$: $P\left(N_{△}\left(G_{\left(m\right)}\right)<(1-\delta )E\left(N_{△}\left(G_{\left(m\right)}\right)\right)\right)=\left(\begin{array}{cc}exp\left(-\Theta \left({\delta }^2{n}^3\right)\right)& \text{if}{n}^{-3/2}\ll \delta \ll n^{-1}\\ exp\left(-\Theta \left({\delta }^{2/3}{n}^2\right)\right)& \text{if}{n}^{-3/4}\ll \delta \ll 1.\end{array}\right)$ Neeman et al. (2023) also conjectured that the probability should be of the form $exp\left(-\Theta \left({\delta }^2{n}^3\right)\right)$ in the “missing interval” $n^{-1}\ll \delta \ll n^{-3/4}$. We prove this conjecture. As part of our proof we also prove that some random graph statistics, related to degrees and codegrees, are normally distributed with high probability.