Extremes of log-correlated Gaussian field

Abstract

Extremes of log-correlated Gaussian fieldsExtreme value 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 finding an appropriate set of assumptions on the covariance structure at microscopic and macroscopic levels which are good enough to ensure convergence of distribution of the maximum, after centering.