A local bias approach to the clustering of discrete density peaks
  Maxima of the linear density field form a point process that can be used to
understand the spatial distribution of virialized halos that collapsed from
initially overdense regions. However, owing to the peak constraint, clustering
statistics of discrete density peaks are difficult to evaluate. For this
reason, local bias schemes have received considerably more attention in the
literature thus far. In this paper, we show that the 2-point correlation
function of maxima of a homogeneous and isotropic Gaussian random field can be
thought of, up to second order at least, as arising from a local bias expansion
formulated in terms of rotationally invariant variables. This expansion relies
on a unique smoothing scale, which is the Lagrangian radius of dark matter
halos. The great advantage of this local bias approach is that it circumvents
the difficult computation of joint probability distributions. We demonstrate
that the bias factors associated with these rotational invariants can be
computed using a peak-background split argument, in which the background
perturbation shifts the corresponding probability distribution functions.
Consequently, the bias factors are orthogonal polynomials averaged over those
spatial locations that satisfy the peak constraint. In particular, asphericity
in the peak profile contributes to the clustering at quadratic and higher
order, with bias factors given by generalized Laguerre polynomials. We
speculate that our approach remains valid at all orders, and that it can be
extended to describe clustering statistics of any point process of a Gaussian
random field. Our results will be very useful to model the clustering of
discrete tracers with more realistic collapse prescriptions involving the tidal
shear for instance.
