We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and an estimate for the right tail up to a multiplicative constant on the law of the centered maximum.
"Extreme values for two-dimensional discrete Gaussian free field." Ann. Probab. 42 (4) 1480 - 1515, July 2014. https://doi.org/10.1214/13-AOP859