论文标题
单色子图的渐近正态性的第四刻现象
A fourth moment phenomenon for asymptotic normality of monochromatic subgraphs
论文作者
论文摘要
给定图形序列$ \ {g_n \} _ {n \ ge1} $和一个简单的连接子图$ h $，我们用$ t（h，g_n）$表示单色副本的数量，$ h $的单色副本数量，均匀随机的$ g_n $ a $ g_n $ a $ c \ ge ge 2 $ 2 $ 2 $ 2 $ $ g_n $。在本文中，我们证明了具有明确错误率的$ t（h，g_n）$的中心限制定理。错误率是由通过加入$ h $的副本形成的收集的图计数而引起的，我们称之为良好的加入。良好连接的计数与$ t（h，g_ {n}）$的归一化版本的第四刻密切相关，并且该连接使我们能够显示中心限制定理的第四刻现象。 确切地说，对于$ c \ ge 30 $，我们表明$ t（h，g_n）$（适当居中和重新定制）分配时会收敛到$ \ nathcal {n}（0,1）$，每当它的第四刻收敛到3（标准正态分布的第四次）时，我们表明，在$ c \ ge 2 $时获得正常限制是必要的第四刻的融合。这些结果的组合表明，第四刻的条件是所有子图$ h $（只要$ c \ ge 30 $，$ t（h，g_n）$的限制正态分布（h，g_n）$。
Given a graph sequence $\{G_n\}_{n\ge1}$ and a simple connected subgraph $H$, we denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \ge 2$ colors. In this article, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of $H$ that we call good joins. Counts of good joins are closely related to the fourth moment of a normalized version of $T(H,G_{n})$, and that connection allows us to show a fourth moment phenomenon for the central limit theorem. Precisely, for $c\ge 30$, we show that $T(H,G_n)$ (appropriately centered and rescaled) converges in distribution to $\mathcal{N}(0,1)$ whenever its fourth moment converges to 3 (the fourth moment of the standard normal distribution). We show the convergence of the fourth moment is necessary to obtain a normal limit when $c\ge 2$. The combination of these results implies that the fourth moment condition characterizes the limiting normal distribution of $T(H,G_n)$ for all subgraphs $H$, whenever $c\ge 30$.
