Local central limit theorem for triangle counts in sparse random graphs

Abstract Let $X_H$ be the number of copies of a fixed graph H in G ( n , p ). In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as H is connected, $p\gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$ , where m ( H ) denotes the m -density of H . Recently, Sah and Sawhney showed that the Gilmer–Kopparty conjecture holds for constant p . In this paper, we show that the Gilmer–Kopparty conjecture holds for triangle counts in the sparse range. More precisely, if $p \in (4n^{-1/2}, 1/2)$ , then \[ {} {} {}\sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|=n^{-1/2+o(1)}p^{1/2}, {} {}\] where $\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$ , $X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$ and $\mathcal{L}$ is the support of $X^*$ . By combining our result with the results of Röllin–Ross and Gilmer–Kopparty, this establishes the Gilmer–Kopparty conjecture for triangle counts for $n^{-1}\ll p \lt c$ , for any constant $c\in (0,1)$ . Our quantitative result is enough to prove that the triangle counts converge to an associated normal distribution also in the $\ell_1$ -distance. This is the first local central limit theorem for subgraph counts above the so-called $m_2$ -density threshold.