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

Abstract

In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) 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 and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.Comment: 30 pages, 4 figures. The argument in Lemma 3.1 and 3.4 was revised. Lemma A.4, A.5 and A.6 were added for this reason. Other typos were corrected throughout the article. The proof of Lemma A.1 and A.3 was simplifie