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In this paper, we study a random field constructed from the two-dimensionalGaussian free field (GFF) by modifying the variance along the scales in theneighborhood of each point. The construction can be seen as a local martingaletransform and is akin to the time-inhomogeneous branching random walk. In thecase where the variance takes finitely many values, we compute the first orderof the maximum and the log-number of high points. These quantities wereobtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) whenthe variance is constant on all scales. The proof relies on a truncated secondmoment method proposed by Kistler (2015), which streamlines the proof of theprevious results. We also discuss possible extensions of the construction tothe continuous GFF.

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