Persistent Betti numbers of random \v{C}ech complexes

TL;DR

This paper investigates the asymptotic behavior of persistent Betti numbers in random cech complexes, extending previous results on ordinary Betti numbers to the persistent homology setting in the subcritical regime.

Contribution

It introduces a theoretical framework for analyzing persistent Betti numbers of random cech complexes and determines their asymptotic order, extending prior work on ordinary Betti numbers.

Findings

Determines the asymptotic order of the kth persistent Betti number in the subcritical regime.

Extends Kahle's results from ordinary Betti numbers to persistent Betti numbers.

Provides a new framework for studying persistent homology in random geometric complexes.

Abstract

We study the persistent homology of random \v{C}ech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the kth persistent Betti number of a random \v{C}ech complex and determine its asymptotic order in the subcritical regime. This extends a result of Kahle on the asymptotic order of the ordinary kth Betti number of such complexes to the persistent setting.