Recursions and tightness for the maximum of the discrete, two dimensional Gaussian free field

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 F. M. Dekking and B. Host [Probab. Theory Relat. Fields 90, No. 3, 403–426 (1991; Zbl 0734.60074)] for branching random walks with bounded increments and on comparison results specific to Gaussian fields.