Stochastic Homology of Gaussian vs. non-Gaussian Random Fields: Graphs
  towards Betti Numbers and Persistence Diagrams
  The topology and geometry of random fields - in terms of the Euler
characteristic and the Minkowski functionals - has received a lot of attention
in the context of the Cosmic Microwave Background (CMB), as the detection of
primordial non-Gaussianities would form a valuable clue on the physics of the
early Universe. The virtue of both the Euler characteristic and the Minkowski
functionals in general, lies in the fact that there exist closed form
expressions for their expectation values for Gaussian random fields. However,
the Euler characteristic and Minkowski functionals are summarizing
characteristics of topology and geometry. Considerably more topological
information is contained in the homology of the random field, as it completely
describes the creation, merging and disappearance of topological features in
superlevel set filtrations.
  In the present study we extend the topological analysis of the superlevel set
filtrations of two-dimensional Gaussian random fields by analysing the
statistical properties of the Betti numbers - counting the number of connected
components and loops - and the persistence diagrams - describing the creation
and mergers of homological features. Using the link between homology and the
critical points of a function - as illustrated by the Morse-Smale complex - we
derive a one-parameter fitting formula for the expectation value of the Betti
numbers and forward this formalism to the persistent diagrams. We, moreover,
numerically demonstrate the sensitivity of the Betti numbers and persistence
diagrams to the presence of non-Gaussianities.
