Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the two-dimensional Gaussian free field, are conjectured to form universality class of extreme value statistics (notably in the work of Carpentier & Le Doussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on the two-dimensional discrete Gaussian free field. At low temperature, we show that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. (with L.-P. Arguin, 2015).

In a second work (with M. Pain, 2021), we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

Finally, I will discuss in detail (depending on the time left) recent works with B. Bonnefont (ex-PhD student, now Post-Doc at University of Geneva) and M. Pain (CR @Toulouse), on questions suggested by B. Derrida.