  We consider a homogeneously pumped photon gas coupled to a dye medium and
investigate how its steady state is effected when varying the pump power, the
photon cavity lifetime and the cutoff frequency. We study how the interplay
between pumping, loss, and dye-induced thermalization influences the selection
of the cavity modes that acquire large occupation. Depending on the parameter
regime, the latter can be related either to lasing of (typically multiple)
modes or to equilibrium-like photon condensation in the ground mode. We
calculate and explain the phase diagram of the system, with a particular
emphasis on the role played by a mode repulsion that occurs in the regime of
weak cavity loss.
