稀疏随机图中团计数的局部中心极限定理
A Local Central Limit Theorem for Clique Counts in Sparse Random Graphs
AI总结：
本文针对稀疏随机图$G_{n,p}$，证明固定阶数$r \geq 3$的完全图$K_r$的副本计数满足局部中心极限定理，基本解决了团计数的局部中心极限定理猜想。
AI中文摘要：
设$X_H$为$G_{n,p}$中固定图$H$的副本数量，Gilmer与Kopparty猜想：当$H$连通、$p \gg n^{-1/m(H)}$且$n^2(1-p) \gg 1$时，$X_H$满足局部中心极限定理（LCLT），其中$m(H)$为最大密度。Berkowitz的工作之后，Sah与Sawhney证实了该猜想对所有常数$p$成立，但留下$p=o(1)$的情形未解决。在该情形下，文献中仅处理了$H=K_3$的情况：近期Araújo与Mattos证实$p \in (4n^{-1/2},1/2)$时猜想成立，结合Röllin与Ross的一般结果，基本解决了三角形的猜想。本文将这些结果推广，证明$H=K_r$（任意固定$r \geq 3$）在$n^{-1/m(H)} \ll p \leq 1/2$的范围内满足局部中心极限定理，基本解决了团的猜想。
英文摘要：
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. 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 $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sawhney confirmed this conjecture for every constant $p$, leaving the regime where $p=o(1)$ open. In this regime, the only case addressed in the literature is when $H=K_3$, where, in a recent paper, Araújo and Mattos confirmed the conjecture for $p \in (4n^{-1/2}, 1/2)$. This, together with a general result of Röllin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for $H = K_r$ (for any fixed $r \ge 3$) in the regime $n^{-1/m(H)}\ll p\leq 1/2$, essentially settling the conjecture for cliques.
