Maximally persistent cycles in random geometric complexes

We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degreek 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 ≥ 2 and 1 ≤ k ≤ d - 1 the maximally persistent cycle has (multiplicative) persistence of order {equation presented} with high probability, characterizing its rate of growth as n←∞. The implied constants depend on k, d and on whether we consider the Vietoris-Rips or Čech filtration.