This paper proves that in a wide sparse range, the number of r‑vertex cliques (complete subgraphs on r vertices) in a random graph behaves like a discrete Gaussian. More precisely, when the edge probability p is larger than the known appearance threshold for r‑cliques but at most 1/2, the probability that the clique count equals a particular number is asymptotically the same as the corresponding value of a normal (bell‑curve) density. In short, the authors establish a local central limit theorem for K_r in G_{n,p}, thereby essentially settling a conjecture of Gilmer and Kopparty for cliques.

A few words of background. G_{n,p} is the standard random graph on n labeled vertices where each edge is present independently with probability p. For a fixed small graph H we write X_H for the number of labeled copies of H in G_{n,p}. There are known threshold values of p that control whether X_H is typically zero, Poisson, or satisfies a central limit theorem (CLT). A CLT describes the cumulative distribution of the normalized count, while a local CLT (LCLT) is stronger: it gives the actual point probabilities P(X_H = x) and says they match the Gaussian density at scale 1/σ, where σ is the standard deviation. For an r‑clique K_r the relevant density parameter is m(K_r) = (r−1)/2, and the result applies when p ≫ n^{−1/m(K_r)} and p ≤ 1/2.

What the authors prove and how sharp it is. Their main theorem (Theorem 1.1) shows that, after centering and scaling, the atom probabilities of the r‑clique count equal the corresponding Gaussian density up to a vanishing error. They state this both qualitatively (the error goes to zero) and with explicit quantitative error bounds in several regimes of p. The different error estimates depend on how p compares to another technical threshold called the 2‑density (denoted m_2) and on some polylogarithmic factors. The authors note they did not push to optimize these quantitative bounds; some polynomial factors in n can likely be improved to polylogarithmic terms.