Extremes of the two-dimensional Gaussian free field with scale-dependent variance

In this paper, we study a random field constructed from the two-dimensional Gaussian free field by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel & Giacomin and Daviaud in the case where the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler, which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous Gaussian free field.