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 Čech 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 $d\ge2$ and $1\le k\le d-1$ the maximally persistent cycle has (multiplicative) persistence of order

\[\Theta ((\frac{\log n}{\log\log n})^{1/k}),\] 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 Čech filtration.