  When Peccei-Quinn (PQ) symmetry breaking happens after inflation, the axion
field takes random values in causally disconnected regions. This leads to
fluctuations of order one in the axion energy density around the QCD epoch.
These over-densities eventually decouple from the Hubble expansion and form
so-called miniclusters. We present a semi-analytical method to calculate the
average axion energy density, as well as the power spectrum, from the
re-alignment mechanism in this scenario. Furthermore, we develop a modified
Press & Schechter approach, suitable to describe the collapse of non-linear
density fluctuations during radiation domination, which is relevant for the
formation of axion miniclusters. It allows us to calculate the double
differential distribution of gravitationally collapsed miniclusters as a
function of their mass and size. For instance, assuming a PQ scale of $10^{11}$
GeV, minicluster masses range from about $5 \times 10^{-16}$ to $3 \times
10^{-13}$ solar masses and have sizes from about $4\times 10^4$ to $7\times
10^5$ km at the time they start to collapse.
