Local central limit theorem for triangle counts in sparse random graphs

Let XH 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 XH as long as H is connected, p n-1/m(H) and n²(1-p) 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 ∈ (4n-1/2, 1/2) , then \[ {} {} {}x∈ L| {1}{√2π}e-x^2/2-σ· P(X^* = x)|=n-1/2+o(1)p1/2, {} {}\] where σ² = Var(XK₃) , X*=(XK₃-E(XK₃))/σ and 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 p < c , for any constant c∈ (0,1) . Our quantitative result is enough to prove that the triangle counts converge to an associated normal distribution also in the ₁ -distance. This is the first local central limit theorem for subgraph counts above the so-called m₂ -density threshold.