  We revisit the magnetic properties of the hexagonal ferrite
PbFe$_{12-x}$Ga$_x$O$_{19}$. Recent experiments have reported puzzling
dependencies of the ordering temperature and the saturation magnetization on
the Ga concentration $x$. To explain these observations, we perform large-scale
Monte Carlo simulations, focusing on the effects of an unequal distribution of
the Ga impurities over the five distinct Fe sublattices. Ab-initio
density-functional calculations predict that the Ga ions preferably occupy the
$12k$ sublattice and (to a lesser extent) the $2a$ sublattice. We incorporate
this insight into a nonuniform model of the Ga distribution. Monte Carlo
simulations using this model lead to an excellent agreement between the
theoretical and experimental values of the ordering temperature and saturation
magnetization, indicating that the unequal distribution of the Ga impurities is
the main reason for the unusual magnetic properties of
PbFe$_{12-x}$Ga$_x$O$_{19}$. We also compute the temperature and concentration
dependencies of the sublattice magnetizations, and we study the character of
the zero-temperature transition that takes place when the ordering temperature
is tuned to zero.
