Maximally Persistent Cycles in Random Geometric Complexes

Abstract

We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the \cech or the Vietoris--Rips filtration built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1{]}^d$. This is a natural way of measuring the largest "$k$-dimensional hole" in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all $d\ge 2$ and $1\le k\le d-1$ the maximally persistent cycle has (multiplicative) persistence of order $$\Theta \left({\left(\frac{\log n}{\log \log n}\right)}^{1/k}\right),$$ with high probability, characterizing its rate of growth as $n\to \infty$. The implied constants depend on $k$, $d$, and on whether we consider the Vietoris--Rips or \cech filtration.

revised according to referee reports. 35 pages, 7 figures