Talk information

Date: Sunday, June 14, 2026
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Ilay Hoshen (Tel Aviv University)
Title: Local Central Limit Theorem for Clique Counts in $G(n, p)$

Abstract:

Let $X_H$ denote the number of copies of a fixed graph $H$ in $G(n, p)$. In 2016, Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2(1-p) \gg 1$. While a standard CLT describes the convergence of the cumulative distribution function, an LCLT provides a much sharper estimate: it shows that the probability of $X_H$ taking any specific integer value $k$ is asymptotically equal to the density of the corresponding normalised Gaussian distribution at $k$.

Following the above, Sah and Sawhney (2020) confirmed the conjecture for constant $p$, leaving the regime where $p=o(1)$ open. In this regime, the only case addressed in the literature is when $H=K_3$. Indeed, in a recent paper, Araújo and Letícia (2023) confirmed the conjecture for $H = K_3$ and $p \in (4n^{-1/2}, 1/2)$, which, together with a general result of Röllin and Ross, settles the conjecture for triangles. In this talk, we present an extension of this result, and show that an LCLT holds for $H = K_r$ (for any fixed $r \ge 3$) in the regime $p \in (C n^{-1/m_2(H)}, 1/2)$.

This is joint work with Asaf Cohen Antonir and Maksim Zhukovskii.