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 VietorisRips 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 $1 \le k \le d1$ 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 VietorisRips or \cech filtration.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.04347
 Bibcode:
 2015arXiv150904347B
 Keywords:

 Mathematics  Probability;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 Primary: 60B99;
 60D05;
 05E45;
 Secondary: 55U10
 EPrint:
 revised according to referee reports. 35 pages, 7 figures