  The Peak-Patch algorithm is used to identify the densest minicluster seeds in
the initial axion density field simulated from string decay. The fate of these
dense seeds is found by tracking the subsequent gravitational collapse in
cosmological $N$-body simulations. We find that miniclusters at late times are
well described by NFW profiles, although for around 80\% of simulated
miniclusters a single power-law density profile of $r^{-2.9}$ is an equally
good fit due to the unresolved scale radius. Under the assumption that all
miniclusters with an unresolved scale radius are described by a power-law plus
axion star density profile, we identify a significant number of miniclusters
that might be dense enough to give rise to gravitational microlensing if the
axion mass is $0.2 \,\mathrm{meV}\lesssim m_a \lesssim 3\,\mathrm{meV}$. Higher
resolution simulations resolving the inner structure and axion star formation
are necessary to explore this possibility further.
