  This paper is second in the series, following Pranav et al. (2019), focused
on the characterization of geometric and topological properties of 3D Gaussian
random fields. We focus on the formalism of persistent homology, the mainstay
of Topological Data Analysis (TDA), in the context of excursion set formalism.
We also focus on the structure of critical points of stochastic fields, and
their relationship with formation and evolution of structures in the universe.
  The topological background is accompanied by an investigation of Gaussian
field simulations based on the LCDM spectrum, as well as power-law spectra with
varying spectral indices. We present the statistical properties in terms of the
intensity and difference maps constructed from the persistence diagrams, as
well as their distribution functions. We demonstrate that the intensity maps
encapsulate information about the distribution of power across the hierarchies
of structures in more detailed than the Betti numbers or the Euler
characteristic. In particular, the white noise ($n = 0$) case with flat
spectrum stands out as the divide between models with positive and negative
spectral index. It has the highest proportion of low significance features.
This level of information is not available from the geometric Minkowski
functionals or the topological Euler characteristic, or even the Betti numbers,
and demonstrates the usefulness of hierarchical topological methods. Another
important result is the observation that topological characteristics of
Gaussian fields depend on the power spectrum, as opposed to the geometric
measures that are insensitive to the power spectrum characteristics.
