The clustering of critical points in the evolving cosmic web
  Focusing on both small separations and Baryonic Acoustic Oscillation scales,
the cosmic evolution of the clustering properties of peak, void, wall, and
filament-type critical points is measured using two-point correlation functions
in $\Lambda$CDM dark matter simulations as a function of their relative rarity.
A qualitative comparison to the corresponding theory for Gaussian Random fields
allows us to understand the following observed features: i) the appearance of
an exclusion zone at small separation, whose size depends both on rarity and on
the signature (\ie the number of negative eigenvalues) of the critical points
involved; ii) the amplification of the Baryonic Acoustic Oscillation bump with
rarity and its reversal for cross-correlations involving negatively biased
critical points; iii) the orientation-dependent small-separation divergence of
the cross-correlations of peaks and filaments (voids and walls) which reflects
the relative loci of such points in the filament's (wall's) eigenframe. The
most significant features of the correlations are tabulated. The (cross-)
correlations involving the most non-linear critical points (peaks, voids)
display significant variation with redshift, while those involving less
non-linear critical points seem mostly insensitive to redshift evolution, which
should prove advantageous to model. The relative distances to the maxima of the
peak-to-wall and peak-to-void over that of the peak-to-filament
cross-correlation are in ratios of $\sim\sqrt{2}$ and $\sim\sqrt{3}$,
respectively which could be interpreted as an indication of the cosmic crystal
being on average close to a cubic lattice. The insensitivity to redshift
evolution suggests that the absolute and relative clustering of critical points
could become a topologically robust alternative to standard clustering
techniques when analyzing upcoming large scale surveys such as Euclid or LSST.
