Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Extremal process in the weakly correlated regime

We prove convergence of the full extremal process of the two-dimensional scale-inhomogeneous discrete Gaussian free field in the weak correlation regime. The scale-inhomogeneous discrete Gaussian free field is obtained from the 2d discrete Gaussian free field by modifying the variance through a function $\mathcal{I}:[0,1]\to [0,1]$. The limiting process is a cluster Cox process. The random intensity of the Cox process depends on the ${\mathcal{I}}^{\mathrm{\prime }}(0)$ through a random measure $Y$ and on the ${\mathcal{I}}^{\mathrm{\prime }}(1)$ through a constant $\beta$. We describe the cluster process, which only depends on ${\mathcal{I}}^{\mathrm{\prime }}(1)$, as points of a standard 2d discrete Gaussian free field conditioned to be unusually high.