Stochasticity in halo formation and the excursion set approach
  The simplest stochastic halo formation models assume that the traceless part
of the shear field acts to increase the initial overdensity (or decrease the
underdensity) that a protohalo (or protovoid) must have if it is to form by the
present time. Equivalently, it is the difference between the overdensity and
(the square root of the) shear that must be larger than a threshold value. To
estimate the effect this has on halo abundances using the excursion set
approach, we must solve for the first crossing distribution of a barrier of
constant height by the random walks associated with the difference, which is
now (even for Gaussian initial conditions) a non-Gaussian variate. The
correlation properties of such non-Gaussian walks are inherited from those of
the density and the shear, and, since they are independent processes, the
solution is in fact remarkably simple. We show that this provides an easy way
to understand why earlier heuristic arguments about the nature of the solution
worked so well. In addition to modelling halos and voids, this potentially
simplifies models of the abundance and spatial distribution of filaments and
sheets in the cosmic web.
