  Shot noise is an important ingredient to any measurement or theoretical
modeling of discrete tracers of the large scale structure. Recent work has
shown that the shot noise in the halo power spectrum becomes increasingly
sub-Poissonian at high mass. Interestingly, while the halo model predicts a
shot noise power spectrum in qualitative agreement with the data, it leads to
an unphysical white noise in the cross halo-matter and matter power spectrum.
In this work, we show that absorbing all the halo model sources of shot noise
into the halo fluctuation field leads to meaningful predictions for the shot
noise contributions to halo clustering statistics and remove the unphysical
white noise from the cross halo-matter statistics. Our prescription
straightforwardly maps onto the general bias expansion, so that the
renormalized shot noise terms can be expressed as combinations of the halo
model shot noises. Furthermore, we demonstrate that non-Poissonian
contributions are related to volume integrals over correlation functions and
their response to long-wavelength density perturbations. This leads to a new
class of consistency relations for discrete tracers, which appear to be
satisfied by our reformulation of the halo model. We test our theoretical
predictions against measurements of halo shot noise bispectra extracted from a
large suite of numerical simulations. Our model reproduces qualitatively the
observed sub-Poissonian noise, although it underestimates the magnitude of this
effect.
