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 scale-inhomogeneous discrete Gaussian free field in dimension two 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 I : [0, 1] -> [0, 1]. The full extremal process converges to a cluster Cox process. The random intensity of the Cox process depends on I' (0) through a random measure Y and on I' (1) through a constant beta. We show that, in law, the random measure, Y, is equal to the Liouville Quantum Gravity measure at sub-critical temperature f alpha = 2 sigma(0). The cluster process, which only depends on I0 (1), can be described as atoms of a standard 2d discrete Gaussian free field conditioned to be unusually high.