  We study the nonequilibrium steady-state of interacting photons in cavity
arrays as described by the driven-dissipative Bose-Hubbard and spin-$1/2$ XY
model. For this purpose, we develop a self-consistent expansion in the inverse
coordination number of the array ($\sim 1/z$) to solve the Lindblad master
equation of these systems beyond the mean-field approximation. Our formalism is
compared and benchmarked with exact numerical methods for small systems based
on an exact diagonalization of the Liouvillian and a recently developed
corner-space renormalization technique. We then apply this method to obtain
insights beyond mean-field in two particular settings: (i) We show that the
gas--liquid transition in the driven-dissipative Bose-Hubbard model is
characterized by large density fluctuations and bunched photon statistics. (ii)
We study the antibunching--bunching transition of the nearest-neighbor
correlator in the driven-dissipative spin-$1/2$ XY model and provide a simple
explanation of this phenomenon.
