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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 holdfor $X_H$ as long as $H$ is connected, $p \gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$.Recently, Sah and Sahwney showed that the Gilmer--Kopparty conjecture holds forconstant $p$. In this paper, we show that the Gilmer--Kopparty conjecture holdsfor triangle counts in the sparse range. More precisely, there exists $C>0$such that if $p \in (Cn^{-1/2}, 1/2)$, then \[ \sup_{x\in \mathcal{L}}\left|\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* =x)\right|\rightarrow 0,\] 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 supportof $X^*$. By combining our result with the results of R\"ollin--Ross andGilmer--Kopparty, this establishes the Gilmer--Kopparty conjecture for trianglecounts for $n^{-1}\ll p < c$, for any constant $c\in (0,1)$. Our result is thefirst local central limit theorem for subgraph counts above the $m_2$-density.

Local central limit theorem for triangle counts in sparse random graphs