2015/09/14 by Bobrowski, Omer, Kahle, Matthew, Skraba, Primoz · 1 citation
#05E45 #60D05 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Primary: 60B99 #Probability (math.PR) #Secondary: 55U10
paper · doi:10.48550/arxiv.1509.04347
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 ≥ 2 and 1 ≤ k ≤ d-1 the maximally persistent cycle has (multiplicative) persistence of order Θ(((log n)/(log log n) )1/k ), 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 \cech filtration.