The maximal displacement of the two dimensional Gaussian free field has been a topic of substantial interest recently. For a family of Gaussian random variables in a square of side N with covariance structure given by the Gaussian free field, what can be said about the distribution of the maximum of these variables as N goes to infinity? For instance, is this distribution tight after centering about the mean? We answer this question in the affirmative by locating the distribution to within a bounded error. The reasoning relies strongly on techniques from branching Brownian motion that have been known since the 1970s. This talk is based on joint work with O. Zeitouni.

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