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 \& Ledoussal 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. (Joint work with L.-P. Arguin).
Biography
Olivier Zindy graduated at the École Normale Supérieure in Cachan before receiving his Ph.D. in mathematics at the University Paris 6, France, in 2007. After a postdoc position in Berlin, Germany, he moved in 2009 to the University Paris 6 as an assistant professor. His main research interests focus on random walks in random environment, branching processes and log-correlated Gaussian fields.