  We study the small scale structure of axion dark matter in the
post-inflationary scenario, which predicts the formation of low-mass,
high-density clumps of gravitationally bound axions called axion miniclusters.
To this end, we follow numerically the cosmological evolution of the axion
field and the network of strings and domain walls until the density contrast is
frozen. Our simulations, comprising up to $8192^3$ points, are the largest
studies of the axion field evolution in the non-linear regime presented so far.
Axitons, pseudo-breathers of the Klein-Gordon equation, are observed to form in
our simulation at late times. Studying their properties analytically and
numerically, we observe that in particular, the earliest axitons contribute to
density perturbations at the typical length scale of miniclusters. We analyse
the small scale structure of the density field, giving the correlation length,
power spectrum and the distribution of high-density regions that will collapse
into axion miniclusters. The final density field of our simulations can be used
to calculate the minicluster mass fraction in simulations including gravity. In
particular, we find that typical minicluster progenitors are smaller than
previously thought and only of moderate, $\mathcal{O}(1)$ overdensity. We
expect these miniclusters to have a rich sub-structure, emerging from
small-scale fluctuations produced in the collapse of the string-wall network
and from axitons.
