发表机构
Institute of Science and Technology Austria (ISTA); University of Chicago（奥地利科学技术研究所（ISTA）; 芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出非线性下尾大偏差的通用方法，填补随机图三角形缺失概率在临界状态的空白，并应用于设计理论，获得相关计数紧估计。
AI 中文摘要
我们引入了通用方法，用于推导多种组合学感兴趣的非线性问题的下尾大偏差原理。这些方法在“临界”状态下尤其有效，此时非线性不够稀疏，无法应用泊松下尾（例如，Janson不等式不能提供尖锐的尾部界限），但情况也不至于密集到组合效应占主导（例如，我们不能直接应用超图容器理论或Kozma-Samotij的相对熵框架）。特别是，这些状态是预期会出现有趣的相变的状态。我们的结果对随机图中的子图有许多推论，回答了Warnke和Jenssen-Perkins-Potukuchi-Simkin的各种问题。例如，考虑随机图$G\sim \mathbb G(n,p)$，并令$L_\triangle(n,p)=\log\Pr[G\text{ 是无三角形的}]$。$L_\triangle(n,p)$的一阶渐近在所有参数范围内早已为人所知，除了临界状态，即$p$的数量级为$1/\sqrt n$。我们能够填补这一空白：对于任意固定的$c>0$，我们证明$n^{-3/2}L_\triangle(n,c/\sqrt n)$收敛到一个由双参数变分问题表示的极限，该问题在$c\approx 4.341$处有一个单一的相变。相反，我们证明对于任何固定的偶圈$C_{2\ell}$，$G$为$C_{2\ell}$-free的概率不存在这样的相变。我们还在组合设计理论中获得了一些应用。针对Glock-Kühn-Lo-Osthus、Kelly和Kwan-Sah-Sawhney-Simkin的猜想，我们获得了关于没有Pasch配置的$n$阶Steiner三元组系统数量以及没有$2\times 2$拉丁子方的$n$阶拉丁方数量的新估计（两者在$\exp(o(n^2))$因子内是紧的）。
英文摘要
We introduce general methods to derive lower tail large deviation principles, for a variety of nonlinear problems of combinatorial interest. These methods are especially effective in "critical" regimes, where the nonlinearities are not sparse enough for Poissonian lower tails (e.g., Janson's inequality does not provide a sharp tail bound), but the situation is not so dense that combinatorial effects dominate (e.g., we cannot directly apply the theory of hypergraph containers or the relative entropy framework of Kozma-Samotij). In particular, these are regimes where one expects interesting phase transitions to occur. Our results have a number of consequences related to subgraphs in random graphs, answering various questions of Warnke and Jenssen-Perkins-Potukuchi-Simkin. For example, consider a random graph $G\sim \mathbb G(n,p)$, and let $L_\triangle(n,p)=\log\Pr[G\text{ is triangle-free}]$. The first-order asymptotics of $L_\triangle(n,p)$ have long been known in all parameter ranges except the critical regime where $p$ has order of magnitude $1/\sqrt n$. We are able to fill this gap: for any fixed $c>0$, we show that $n^{-3/2}L_\triangle(n,c/\sqrt n)$ converges to a limit expressed in terms of a two-parameter variational problem, which has a single phase transition at $c\approx 4.341$. In contrast, we show that there is no such phase transition for the probability that $G$ is $C_{2\ell}$-free, for any fixed even cycle $C_{2\ell}$. We also obtain some applications in combinatorial design theory. Addressing conjectures of Glock-Kühn-Lo-Osthus, Kelly, and Kwan-Sah-Sawhney-Simkin, we obtain new estimates on the number of order-$n$ Steiner triple systems with no Pasch configuration and the number of order-$n$ Latin squares with no $2\times 2$ Latin subsquare (both tight up to a factor of $\exp(o(n^2))$).
