发表机构
Institute for Basic Science（基础科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机图中团因子缺失的指数尾部概率，发展可迭代单根Johansson-Kahn-Vu方法，证明最大匹配数的中心极限定理，并解决Surya-Warnke猜想，得到稠密随机图色数的高斯极限。
AI 中文摘要
Johansson、Kahn和Vu的著名结果确定了随机图中团因子的阈值阶，随后的工作确定了尖锐阈值和相应的击中时间现象。本文研究在阈值之上不存在$K_r$-因子的概率，更一般地，研究最大的$K_r$-匹配覆盖少于$G(n,p)$的$n-s$个顶点的概率。对于每个固定的$r\ge3$，在整个范围$$n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},\qquad n-s\in r\mathbb Z,\qquad s=o(n),$$内，我们证明$$\mathbb P\bigl(\phi_r^s(G(n,p))=0\bigr)=\exp\left(-\Theta_r\\!\left((s+1)\frac{\mu_r(n,p)}n\right)\right),$$其中$\phi_r^s(G)$是恰好覆盖$n-s$个顶点的$K_r$-匹配的数量，$\mu_r(n,p):=\binom nrp^{\binom r2}$。下界由$s+1$个不在任何$K_r$副本中的顶点给出。对于上界，我们发展了Johansson--Kahn--Vu方法的一个可迭代的单根版本。作为结构推论，我们证明在整个稀疏团窗口内，$G(n,p)$在每一个极大$K_r$-匹配之外的剩余部分具有一个几乎完美的$K_{r-1}$-匹配。独立地，我们证明了最大$K_r$-匹配数的中心极限定理。结合这些输入和我们早期工作中关于$r=2$的结构定理，我们证明了非常稠密随机图的色数的中心极限定理：对于每个$r\ge2$和$n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},$ $$\frac{\chi(G(n,1-p))-\mathbb E\chi(G(n,1-p))}{\sqrt{\mu_{r+1}(n,p)}/r}\xrightarrow{\mathrm d}\mathcal N(0,1),\qquad\operatorname{Var}\bigl(\chi(G(n,1-p))\bigr) \sim\frac{\mu_{r+1}(n,p)}{r^2}.$$这解决了Surya--Warnke猜想在$r\ge2$的每个团窗口内部的情况，将其浓度预测加强为具有渐近精确方差的高斯极限。
英文摘要
The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more generally, the probability that the largest $K_r$-matching covers less than $n-s$ vertices of $G(n,p)$. For every fixed $r\ge3$, throughout the range $$n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},\qquad n-s\in r\mathbb Z,\qquad s=o(n),$$ we prove $$\mathbb P\bigl(ϕ_r^s(G(n,p))=0\bigr)=\exp\left(-Θ_r\!\left((s+1)\frac{μ_r(n,p)}n\right)\right),$$ where $ϕ_r^s(G)$ is the number of $K_r$-matchings covering exactly $n-s$ vertices and $μ_r(n,p):=\binom nrp^{\binom r2}$. The lower bound is given by $s+1$ vertices which lie in no copy of $K_r$. For the upper bound we develop an iterable one-root version of the Johansson--Kahn--Vu method. As a structural consequence, we show that the remainder of $G(n,p)$ outside every maximal $K_r$-matching has an almost-perfect $K_{r-1}$-matching throughout the sparse clique window. Independently, we prove a central limit theorem for the maximum $K_r$-matching number. Combining these inputs and a structural theorem for $r=2$ from our earlier work, we prove a central limit theorem for the chromatic number of very dense random graphs: for every $r\ge2$ and $n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},$ $$\frac{χ(G(n,1-p))-\mathbb Eχ(G(n,1-p))}{\sqrt{μ_{r+1}(n,p)}/r}\xrightarrow{\mathrm d}\mathcal N(0,1),\qquad\operatorname{Var}\bigl(χ(G(n,1-p))\bigr) \sim\frac{μ_{r+1}(n,p)}{r^2}.$$ This settles the Surya--Warnke conjecture throughout the interior of every clique window with $r\ge2$, strengthening its concentration prediction to a Gaussian limit with asymptotically exact variance.
Comments39 pages
