Hostname: page-component-89b8bd64d-j4x9h Total loading time: 0 Render date: 2026-05-09T07:00:42.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Local central limit theorem for triangle counts in sparse random graphs

Part of: Graph theory Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  14 November 2025

PEDRO ARAÚJO  and
LETÍCIA MATTOS
PEDRO ARAÚJO
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00, Prague, Czech Republic. e-mail: pedro.araujo@cvut.cz
LETÍCIA MATTOS
Affiliation:
Institut für Informatik, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany. e-mail: mattos@uni-heidelberg.de
Get access Rights & Permissions [Opens in a new window]

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.

MSC classification

Primary: 05C80: Random graphs
Secondary: 60E10: Characteristic functions; other transforms 60F15: Strong theorems

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 180 , Issue 2 , March 2026 , pp. 459 - 475
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Barbour, A. D.. Poisson convergence and random graphs. Math. Proc. Camb. Phil. Soc., 92(2) (1982), 349–359.CrossRefGoogle Scholar
Barbour, A. D., Karoński, M. and Ruciński, A.. A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory. Ser. B, 47(2) (1989), 125–145.CrossRefGoogle Scholar
Berkowitz, R.. A quantitative local limit theorem for triangles in random graphs. Preprint: ArXiv:1610.01281 (2016).Google Scholar
Berkowitz, R.. A local limit theorem for cliques in G(n, p). Preprint: ArXiv:1811.03527 (2018).Google Scholar
Durrett, R.. Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, (2019).Google Scholar
Erdős, P. and Rényi, A.. On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17–61.Google Scholar
Feller, W.. An Introduction to Probability Theory and its Applications, Volume 2, vol. 81 (John Wiley and Sons, 1991).Google Scholar
Fox, J., Kwan, M. and Sauermann, L.. Anti-concentration for subgraph counts in random graphs. Ann. Probab., 49(3) (2021), 1515–1553.Google Scholar
Fox, J., Kwan, M. and Sauermann, L.. Combinatorial anticoncentration inequalities, with applications. Math. Proc. Camb. Phil. Soc., 171(2) (2021), 227–248.CrossRefGoogle Scholar
Frieze, A. and Karoński, M.. Introduction to Random Graphs (Cambridge University Press, 2016).Google Scholar
Gilmer, J. and Kopparty, S.. A local central limit theorem for triangles in a random graph. Random Structures Algorithms, 48(4) (2016), 732–750.CrossRefGoogle Scholar
Janson, S.. Orthogonal decompositions and functional limit theorems for random graph statistics, vol. 534 (American Mathematical Soc., 1994).CrossRefGoogle Scholar
Janson, S. and Ruciński, A.. The infamous upper tail. Random Structures Algorithms, 20(3) (2002), 317–342.CrossRefGoogle Scholar
Karoński, M. and Ruciński, A.. On the number of strictly balanced subgraphs of a random graph. Graph Theory (1983), 79–83.Google Scholar
Kim, J. H. and Vu, V. H.. Concentration of multivariate polynomials and its applications. Combinatorica 20(3) (2000), 417–434.CrossRefGoogle Scholar
Nowicki, K. and Wierman, J. C.. Subgraph counts in random graphs using incomplete U-statistics methods. Discrete Math., 72(1-3) (1988), 299–310.CrossRefGoogle Scholar
Paley, R. E. and Zygmund, A.. A note on analytic functions in the unit circle. Math. Proc. Camb. Phil. Soc., 28(3) (1932), 266–272.CrossRefGoogle Scholar
Röllin, A. and Ross, N.. Local limit theorems via Landau–Kolmogorov inequalities. Bernoulli 21(2) (2015), 851–880.Google Scholar
Ross, N.. Fundamentals of Stein’s Method vol. 8. (2011).Google Scholar
Ruciński, A.. When are small subgraphs of a random graph normally distributed? Probab. Theory Related Fields, 78(1) (1988), 1–10.CrossRefGoogle Scholar
Sah, A. and Sawhney, M.. Local limit theorems for subgraph counts. J. London Math. Soc., 105(2) (2022), 950–1011.CrossRefGoogle Scholar