Extreme values of correlated Gaussian fields: the spin glass perspective

The study of extremal values of a large collection of random variables dates back to the early 20th century and has been well established in the case of independent or weakly correlated variables. On the other hand, few universality results are known in the case where the random variables are strongly correlated. In the early 1980's, statistical physicists (with notable contributions by Giorgio Parisi) have proposed a compelling universal picture to understand the extremal values of correlated variables for a broad class of models. This picture was largely inspired from the study of spin glass models in physics. In this talk, I will describe the statistical physics approach in the case of Gaussian fields and survey recent rigorous results in establishing this picture. A particular emphasis will be put on the Sherrington-Kirkpatrick model and log-correlated Gaussian fields such as branching Brownian motion and the 2D Gaussian free field.