Density fields and halo mass functions in the Geometrical Adhesion toy
  Model
  In dimension 2 and above, the Burgers dynamics, the so-called "adhesion
model" in cosmology, can actually give rise to several dynamics in the inviscid
limit. We investigate here the statistical properties of the density field when
it is defined by a "geometrical model" associated with this Burgers velocity
field and where the matter distribution is fully determined, at each time step,
by geometrical constructions. Our investigations are based on a set of
numerical experiments that make use of an improved algorithm, for which the
geometrical constructions are efficient and robust. In this work we focus on
Gaussian initial conditions with power-law power spectra of slope $n$ in the
range $-3<n<1$, where a self-similar evolution develops, and we compute the
behavior of power spectra, density probability distributions and mass
functions. As expected for such dynamics, the density power spectra show
universal high-$k$ tails that are governed by the formation of pointlike
masses. The two other statistical indicators however show the same qualitative
properties as those observed for 3D gravitational clustering. In particular,
the mass functions obey a Press-Schechter like scaling up to a very good
accuracy in 1D, and to a lesser extent in 2D. Our results suggest that the
"geometrical adhesion model", whose solution is fully known at all times,
provides a precious tool to understand some of the statistical constructions
frequently used to study the development of mass halos in gravitational
clustering.
