Extremes of the supercritical Gaussian Free Field

Abstract

We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).

arXiv comment15 pages, 2 figures. Minor typos corrected, changed the proof of Theorem 2 (upper bound)