A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes

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 $k$-cyc