A Universal Law of Large Numbers for Extreme Cycles in Random \v{C}ech Complexes

Abstract

We study the maximal multiplicative persistence of $k$-cycles in random \v{C}ech 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 $\Pi_{k,n}$ 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 $\Pi_{k,n} = \Theta\left( \left(\frac{\log n}{\log\log n}\right)^{1/k}\right)$. 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.