  A class of inflationary scenarios for primordial black hole (PBH) formation
include a small barrier in the slope of the potential. There, the inflaton
slows down, generating an enhancement of primordial perturbations. Moreover,
the background solution overcomes the barrier at a very low speed, and large
backward quantum fluctuations can prevent certain regions from overshooting the
barrier. This leads to localized bubbles where the field remains trapped behind
the barrier. In such models, therefore, we have two distinct channels for PBH
production: the standard adiabatic density perturbation channel and the bubble
channel. Here, we perform numerical simulations of bubble formation, addressing
the issues of initial conditions, critical amplitude and bubble expansion.
Further, we explore the scaling behaviour of the co-moving size of bubbles with
the initial amplitude of the field fluctuation. We find that for small to
moderate non-Gaussianity $f_{\rm NL}\lesssim 2.6$, the threshold for the
formation of vacuum bubbles agrees with previous analytical estimates
arXiv:1908.11357 to $5\%$ accuracy or so. We also show that the mass
distribution for the two channels is different, leading to a slightly broader
range of PBH masses. The bubble channel is subdominant for small $f_{\rm NL}$,
and becomes dominant for $f_{\rm NL}\gtrsim 2.6$. We find that the mass of PBHs
in the bubble channel is determined by an adiabatic overdensity surrounding the
bubble at the end of inflation. Remarkably, the profile of this overdensity
turns out to be of type-II. This represents a first clear example showing that
overdensities of type-II can be dominant relative to the standard type-I. We
also comment on exponential tails and on the fact that in models with local
type non-Gaussianity (such as the one considered here), the occurrence of
alternative channels can easily be inferred from unitarity considerations.
