On the inadequacy of N-point correlation functions to describe nonlinear
  cosmological fields: explicit examples and connection to simulations
  Motivated by recent results on lognormal statistics showing that the moment
hierarchy of a lognormal variable completely fails at capturing its information
content in the large variance regime, we discuss in this work the inadequacy of
the hierarchy of correlation functions to describe a correlated lognormal
field, which provides a roughly accurate description of the non-linear
cosmological matter density field. We present families of fields having the
same hierarchy of correlation functions than the lognormal field at all orders.
This explicitly demonstrates the little studied though known fact that the
correlation function hierarchy never provides a complete description of a
lognormal field, and that it fails to capture information in the non-linear
regime, where other simple observables are left totally unconstrained. We
discuss why perturbative, Edgeworth-like approaches to statistics in the
non-linear regime, common in cosmology, can never reproduce or predict that
effect, and why it is however generic for tailed fields, hinting at a breakdown
of the perturbation theory based on the field fluctuations. We make a rough but
successful quantitative connection to N-body simulations results, that showed
that the spectrum of the log-density field carries more information than the
spectrum of the field entering the non-linear regime.
