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We apply the Gaussian trajectories approach to the study of the criticalbehavior of two-dimensional dissipative arrays of nonlinear photonic cavities,in presence of two-photon driving and in regimes of sizable loss rates. Inspite of the highly mixed character of the density matrix of this system, thenumerical approach is able to provide precise estimations of the steady-stateexpectation values, even for large lattices made of more than 100 sites. Byperforming a finite-size scaling of the relevant properties of the steadystate, we extrapolate the behavior of the system in the thermodynamic limit andwe show the emergence of a second-order dissipative phase transition, belongingto the universality class of thermal Ising model. This result indicates theoccurrence of a crossover when the loss rate is increased from the weak-losslimit, in which the phase transition belongs to the universality class of thequantum Ising model

Gaussian trajectory approach to dissipative phase transitions: the case of quadratically driven photonic lattices