2023/07/18 by Pedro Araújo, Araújo, Pedro, Letícia Mattos +1 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2307.09446
openalex publication_date 2023/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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 supx∈ L| \dfrac1√(2π)e-x2/2-σ⋅ ℙ(X^* = x)|=n-1/2+o(1)p1/2, where σ2 = \mathbbVar(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 ℓ1-distance. This is the first local central limit theorem for subgraph counts above the so-called m2-density threshold.