  Ultracompact dark matter (DM) minihalos at masses at and below $10^{-12}$
$M_\odot$ arise in axion DM models where the Peccei-Quinn (PQ) symmetry is
broken after inflation. The minihalos arise from density perturbations that are
generated from the non-trivial axion self interactions during and shortly after
the collapse of the axion-string and domain-wall network. We perform
high-resolution simulations of this scenario starting at the epoch before the
PQ phase transition and ending at matter-radiation equality. We characterize
the spectrum of primordial perturbations that are generated and comment on
implications for efforts to detect axion DM. We also measure the DM density at
different simulated masses and argue that the correct DM density is obtained
for $m_a = 25.2 \pm 11.0$ $\mu\mathrm{eV}$.
