随机单纯形树子网络的极限定律
Limit laws of random simplex tree-child networks
浏览论文内容
中文总结 AI 辅助
研究随机单纯形树子网络相关指数和高度的极限定律，通过重新缩放证明其联合分布极限及高度极限定律，建立尾部界限，获得叶子高度轮廓缩放极限，确定固定根、随机顶点和叶子周围的局部极限。
中文摘要 AI 辅助
我们证明，具有n个分类单元的均匀随机单纯形树子网络的较长和较短Sackin指数，在通过$n^{-7/4}$重新缩放后允许联合分布极限。极限分布由布朗游程的泛函描述。我们还确定了通过$n^{-3/4}$重新缩放后的高度极限定律，从而回答了Zhang（2022）的一个问题。此外，我们为高度建立了尖锐的尾部界限，这意味着上述分布极限中所有矩的收敛。我们进一步获得了叶子整个高度轮廓的缩放极限。最后，我们确定了围绕固定根、均匀随机顶点和均匀随机叶子的大型单纯形网络的局部极限。
英文摘要
We prove that the longer and shorter Sackin indices of a uniformly random simplex tree-child network with $n$ taxa admit joint distributional limits after rescaling by $n^{-7/4}$. The limiting distributions are described by functionals of a Brownian excursion. We also identify the limiting law of the height after rescaling by $n^{-3/4}$, thereby answering a question of Zhang~(2022). Moreover, we establish sharp tail bounds for the height, which imply convergence of all moments in the above distributional limits. We further obtain a scaling limit for the entire height profile of the leaves. Finally, we determine the local limits of large simplex networks around the fixed root, a uniformly random vertex, and a uniformly random leaf.
