  In this article we give a general criterion for some dependent Gaussian
models to belong to maximal domain of attraction of Gumbel, following an
application of the Stein-Chen method studied in Arratia et al(1989). We also
show the convergence of the associated point process. As an application, we
show the conditions are satisfied by some of the well-known supercritical
Gaussian interface models, namely, membrane model, massive and massless
discrete Gaussian free field, fractional Gaussian free field.
