Abstract. We consider the maximum of the discrete two dimensional Gaussian free field (GFF) in a box, and prove that its maximum, centered at its mean, is tight, settling a long-standing conjecture. The proof combines a recent observation of [BDZ10] with elements from [Br78] and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two-dimensional torus, are also discussed.
In REF , Bramson and Zeitouni computed the expected maximum of the DGFF up to an order 1 error and concluded tightness of the recentered maximum.
Tightness of the recentered maximum of the two-dimensional discrete Gaussian Free Field
arXiv: Probability
Mathematics