Extremes of log-correlated Gaussian field

Abstract

Extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields are of significant interest and have been a subject of many recent works. Our work is on a general class of Gaussian fields with logarithmic correlations, of which the discrete Gaussian free field in dimension 2 is a particular example. We will first conclude tightness for this field from the correlation structure. It also involves defining a general class of models with some assumptions on the covariance structure at microscopic and macroscopic levels which are good enough to ensure convergence of distribution of the maximum, after appropriate centering.