We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.
Erwin Bolthausen. Jean-Dominique Deuschel. Ofer Zeitouni. "Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field." Electron. Commun. Probab. 16 114 - 119, 2011. https://doi.org/10.1214/ECP.v16-1610