$G_{n,p}$的最大的诱导树大小围绕标准期望阈值的集中性
Concentration of the largest induced tree size of $G_{n,p}$ around the standard expectation threshold
AI总结：
本文研究了二项随机图$G_{n,p}$中最大诱导树大小的集中性，证明在特定$p$范围内，该大小无法围绕标准期望阈值集中。
摘要：
令$T(G)$为图$G$中最大的诱导树的大小，令$G_{n,p}$为二项随机图。Kamaldinov、Skorkin和Zhukovskii证明，当$p$为常数时，$T(G_{n,p})$以高概率等于两个连续值之一。最近，Oropeza将此结果扩展到所有趋于零的$p$，使得$p > n^{-\ rac{e-2}{3e-2} + ε}$，其中$e$是欧拉常数。我们进一步将此结果扩展到所有趋于零的$p$，使得$p \gg n^{-1/2} \ln^{3/2} n$，并且进一步证明，当$p$满足$n^{-1} \ll p \ll n^{-1/2}$时，$T(G_{n,p})$不能在标准期望阈值周围集中。
英文摘要：
Let $T(G)$ be the size of the largest induced tree of $G$, and let $G_{n,p}$ be the binomial random graph. Kamaldinov, Skorkin, and Zhukovskii proved that $T(G_{n,p})$ equals one of two consecutive values with high probability if $p$ is constant, and more recently, Oropeza extended this result to include all vanishing $p$ such that $p > n^{-\frac{e-2}{3e-2} + ε}$, where $e$ is Euler's constant. We further extend this result to all vanishing $p$ such that $p \gg n^{-1/2} \ln^{3/2} n$, and furthermore, we show that, for $p$ such that $n^{-1} \ll p \ll n^{-1/2}, \ T(G_{n,p})$ cannot be concentrated at the standard expectation threshold.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
