  We adopt a diffusive-Gutzwiller approach to investigate a phase transition in
a quadratically driven-dissipative Bose-Hubbard lattice. Diffusive trajectories
may lead to lower average entanglement as compared to jump-like trajectories
and have a natural tendency to approach coherent states, therefore the method
can be less prone to the bias induced by the fully uncorrelated Gutzwiller
ansatz. Averaging over trajectories does lead to classical correlations and
this allows us to address the correlation length of such 2D lattices of open
quantum systems which is the main goal of this work. Under this approximations,
we find negligible correlation length in the low density phase and apparently
unbounded length grow in the high density phase. Additionally, we show that the
effective relaxation times associated to the times scales for synchronisation
in the high density phase may also diverge suggesting the vanishing of the
Lindbladian gap.
