Extremes of log-correlated gaussian fields

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. Similarly works on extreme values of other Gaussian processes have been in focus, which have a similar correlation structure. The talk is on generalizing results on extreme values for a general class of Gaussian fields with logarithmic correlations, of which the Gaussian membrane model at the critical dimension is a particular example. We try to cover as many models as possible within class and in order to do so we make our assumptions as relaxed as possible. It involves defining a general class of models with some assumptions on the covariance structure at microscopic and macroscopic levels. This is the most general class of models having the log-correlated structure and also satisfying the convergence results that we have proved. The talk involves two steps, one showing tightness and another showing full convergence. We make different set of assumptions for the two classes. Also we discuss some results about the geometry of high values of the field.