Lattices of coupled nonlinear optical cavities are attracting a considerable interest for the possibility of creating novel correlated phases of light . When interacting quantum systems are driven out of equilibrium, the competition between coherent effects and losses in the master equation  can lead to dissipative phase transitions in the thermodynamic limit. Here, we investigate the driven-dissipative Bose-Hubbard model describing photonic lattice systems in the regime of interactions which is relevant for semiconductor polaritonic systems. In 2D lattices the system exhibits a critical slowing down of the dynamics for increasing lattice size in the region of parameters where mean-field theory predicts non-linear bistability . Our investigation of the role of disorder  shows which critical features survive in finite size systems.
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 F. Vicentini, F. Minganti, R. Rota, G. Orso and C. Ciuti, Critical slowing down in driven-dissipative Bose-Hubbard lattices, Phys. Rev. A 97, 013853 (2018).
 F. Vicentini, F. Minganti, A. Biella, G. Orso and C. Ciuti, Optimal stochastic unraveling of disordered open quantum systems: Application to driven-dissipative photonic lattices, Phys. Rev. A 99, 032115 (2019)
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