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

Abstract

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 I : [0, 1] → [0, 1]. The limiting process is a cluster Cox process. The random intensity of the Cox process depends on the I ′ (0) through a random measure Y and on the I ′ (1) through a constant β. We describe the cluster process, which only depends on I ′ (1), as points of a standard 2d discrete Gaussian free field conditioned to be unusually high.