  In this paper, we prove a universality result for the limiting distribution
of persistence diagrams arising from geometric filtrations over random point
processes. Specifically, we consider the distribution of the ratio of
persistence values (death/birth), and show that for fixed dimension,
homological degree and filtration type (Cech or Vietoris-Rips), the limiting
distribution is independent of the underlying point process distribution, i.e.,
universal. In proving this result, we present a novel general framework for
universality in scale-invariant functionals on point processes. Finally, we
also provide a number of new results related to Morse theory in random
geometric complexes, which may be of an independent interest.
