  Magnetic properties of Ga$_{1-x}$Mn$_x$N are studied theoretically by
employing a tight binding approach to determine exchange integrals $J_{ij}$
characterizing the coupling between Mn spin pairs located at distances $R_{ij}$
up to the 16th cation coordination sphere in zinc-blende GaN. It is shown that
for a set of experimentally determined input parameters there are no itinerant
carriers and the coupling between localized Mn$^{3+}$ spins in GaN proceeds via
superexchange that is ferromagnetic for all explored $R_{ij}$ values. Extensive
Monte Carlo simulations serve to evaluate the magnitudes of Curie temperature
$T_\mathrm{C}$ by the cumulant crossing method. The theoretical values of
$T_\mathrm{C}(x)$ are in quantitative agreement with the experimental data that
are available for Ga$_{1-x}$Mn$_x$N with randomly distributed Mn$^{3+}$ ions
with the concentrations $0.01 \leq x \leq 0.1$.
