On persistent homology of random \v{C}ech complexes

TL;DR

This paper explores the relationship between critical simplices and persistence diagrams in ech filtrations, providing new insights into their structure and convergence properties in random point processes.

Contribution

It establishes a connection between critical simplices and persistence diagram points, and derives convergence results for ech filtrations over binomial point processes.

Findings

Critical simplices correspond to points in persistence diagrams.

Number of persistence points relates to critical simplices.

Convergence results for ech filtrations on random point sets.

Abstract

The paper studies the relation between critical simplices and persistence diagrams of the \v{C}ech filtration. We show that adding a critical $k$-simplex into the filtration corresponds either to a point in the $k$th persistence diagram or a point in the $(k-1)$st persistence diagram. Consequently, the number of points in persistence diagrams can be expressed in terms of the number of critical simplices. As an application, we establish some convergence results related to persistence diagrams of the \v{C}ech filtrations built over binomial point processes.