Convergence in law of the centered maximum of the mollified Gaussian free field in two dimensions

Abstract

Consider a family of centered Gaussian fields on the d-dimensional unit box, whose covariance decreases logarithmically in the distance between points. In Part I of this thesis, we prove tightness of the centered maximum of the Gaussian fields and provide exponentially decaying bounds on the right and left tails. Part II is devoted to the study of a specific and fundamental example of a log-correlated Gaussian field in two dimensions, namely, the mollified Gaussian free field (MGFF). The MGFF is a random field obtained by suitably mollifying the covariance of the continuum Gaussian free field, which is a generalized random field defined on measures in the unit square. We prove that the centered maximum of the MGFF converges in law as the mollifier approaches the Dirac delta function. We moreover show that this limit law does not depend on the specific mollifier that is employed, and give a representation for it. Our approach in both Part I and II is similar to the approach employed by Bramson, Ding and Zeitouni in their papers on the centered maximum of the discrete Gaussian free field.