A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes
Abstract
We study the maximal multiplicative persistence of k-cycles in random Čech complexes. Let f:ℝd→ℝ be a probability density function, let Pn be a Poisson process with intensity nf, and let Πk,n denote the largest death-to-birth ratio among all non-essential k-cycles (1≤ k ≤ d−1). For the uniform distribution in the unit hypercube, it was proved in [9] that Πk,n = Θ( ((log n)/(loglog n))1/k). 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.
Related
BibTeX
@misc{bobrowski2026universal,
title = {{A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes}},
author = {Omer Bobrowski and Primoz Skraba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19474},
eprint = {2609.19474},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19474},
}
RIS
TY - UNPB
TI - A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes
AU - Omer Bobrowski
AU - Primoz Skraba
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.19474
UR - https://arxiv.org/abs/2609.19474
AB - We study the maximal multiplicative persistence of k-cycles in random Čech complexes. Let f:ℝ^d→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≤ k ≤ d−1). For the uniform distribution in the unit hypercube, it was proved in [9] that Π_k,n = Θ( ((log n)/(loglog n))^1/k). 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.
ER -