Extremal life times of persistent loops and holes

Abstract

Persistent homology captures the appearances and disappearances of topological features such as loops and holes when growing disks centered at a Poisson point process. We study extreme values for the life times of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation. First, we describe the scaling of the minimal life times for general feature dimensions, and of the maximal life times for holes in the \v{C}ech complex. Then, we proceed to a more refined analysis and establish Poisson approximation for large life times of holes and for small life times of loops. Finally, we also study the scaling of minimal life times in the Vietoris-Rips setting and point to a surprising difference to the \v{C}ech complex.