发表机构
Queen Mary University of London; University of Ljubljana（伦敦大学皇家玛丽学院; 卢布尔雅那大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明随机 Čech 复形中极端环的乘法持久性满足大数定律，极限常数仅依赖维度，不依赖分布，揭示普遍性现象。
AI 中文摘要
我们研究随机 Čech 复形中 $k$-环的最大乘法持久性。设 $f:\mathbb{R}^d\to\mathbb{R}$ 为概率密度函数，设 ${P}_n$ 为强度为 $nf$ 的泊松过程，并设 $\Pi_{k,n}$ 表示所有非本质 $k$-环（$1\le k \le d-1$）中最大的死亡-出生比。对于单位超立方体中的均匀分布，文献[9]证明了 $\Pi_{k,n} = \Theta\left( \left(\frac{\log n}{\log\log n}\right)^{1/k}\right)$。在本文中，我们将此结果改进并推广为一大类分布的大数定律。最重要的是，我们证明极限常数仅依赖于 $d$ 和 $k$，而不依赖于概率密度 $f$。因此，乘法持久性的极值表现出普遍性现象。我们证明该极限常数由 $k$ 维球面的渐近覆盖密度决定。我们的证明识别了最大环背后的几何机制，即一个持久的等周不等式，它给出了生成高持久性环所需点数的尖锐界。通过将覆盖密度估计与等周不等式相结合，我们证明该最小值渐近地由 $k$-球面的有效覆盖所达到。一个关键要素是几何测度论论证，它利用紧性将离散覆盖计数与极限环的体积联系起来。
英文摘要
We study the maximal multiplicative persistence of $k$-cycles in random Čech complexes. Let $f:\mathbb{R}^d\to\mathbb{R}$ be a probability density function, let ${P}_n$ be a Poisson process with intensity $nf$, and let $Π_{k,n}$ denote the largest death-to-birth ratio among all non-essential $k$-cycles ($1\le k \le d-1$). For the uniform distribution in the unit hypercube, it was proved in [9] that $Π_{k,n} = Θ\left( \left(\frac{\log n}{\log\log n}\right)^{1/k}\right)$. In this paper we sharpen and extend this result to a law of large numbers, for a broad class of distributions. Most significantly, we show that the limiting constant depends only on $d$ and $k$, and not on the probability density $f$. Thus the extreme value of multiplicative persistence exhibits a universality phenomenon. We show that the limiting constant is determined by the asymptotic covering density of the $k$-dimensional sphere. Our proof identifies the geometric mechanism underlying maximal cycles, a persistent isoperimetric inequality, which gives sharp bounds on the number of points needed to generate a highly persistent cycle. By combining covering-density estimates with isoperimetric inequalities, we show that this minimum is asymptotically attained by efficient coverings of a $k$-sphere. A key ingredient is a geometric measure theory argument that uses compactness to relate discrete covering counts to the volume of a limiting cycle.
