  Topological Data Analysis (TDA) refers to an approach that uses concepts from
algebraic topology to study the "shapes" of datasets. The main focus of this
paper is persistent homology, a ubiquitous tool in TDA. Basing our study on
this, we investigate the topological dynamics of extreme sample clouds
generated by a heavy tail distribution on $\mathbb R^d$. In particular, we
establish various limit theorems for the sum of bar lengths in the persistence
barcode plot, a graphical descriptor of persistent homology. It then turns out
that the growth rate of the sum of the bar lengths and the properties of the
limiting processes all depend on the distance of the region of interest in
$\mathbb R^d$ from the weak core, that is, the area in which random points are
placed sufficiently densely to connect with one another. If the region of
interest becomes sufficiently close to the weak core, the limiting process
involves a new class of Gaussian processes.
