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We prove convergence of the full extremal process of the two-dimensionalscale-inhomogeneous discrete Gaussian free field in the weak correlationregime. The scale-inhomogeneous discrete Gaussian free field is obtained fromthe 2d discrete Gaussian free field by modifying the variance through afunction $\mathcal{I}:[0,1]\rightarrow [0,1]$. The limiting process is acluster Cox process. The random intensity of the Cox process depends on the$\mathcal{I}^\prime(0)$ through a random measure $Y$ and on the$\mathcal{I}^\prime(1)$ through a constant $\beta$. We describe the clusterprocess, which only depends on $\mathcal{I}^\prime(1)$, as points of a standard2d discrete Gaussian free field conditioned to be unusually high.

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