  The tail of the distribution of primordial fluctuations (corresponding to the
likelihood of realization of large fluctuations) is of interest, from both
theoretical and observational perspectives. In particular, it is relevant for
the accurate evaluation of the primordial black hole (PBH) abundance. In this
paper, we first analyze the non-perturbative $\delta N$ formalism as a method
to non-perturbatively estimate the probability distribution function (PDF) of
primordial fluctuations, discuss its underlying assumptions and deal with
several subtleties that may arise as a result of considering large
fluctuations. Next, we employ the method to study several non-attractor
single-field inflationary models as the simplest examples that may lead to the
abundant production of PBHs. We conclude that the Gaussian extrapolation from
linear perturbation theory may fail drastically to predict the likelihood of
large fluctuations. Specifically, we show that a truncation of the tail, a
power-law tail, a double-exponential tail, and a doubly peaked distribution can
all be realized for the curvature perturbation in the single-field
non-attractor models of inflation. We thus show that there is a diverse zoo of
possible tails from inflation so that a model-dependent, non-perturbative study
of the distribution of the primordial fluctuations seems inevitable concerning
PBH abundance.
