Extreme Values of Highly Correlated Gaussian processes: a study of spin glasses and related models

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Understanding the statistics of large values in stochastic processes and their universality is a long-standing goal of probability theory with direct applications in diverse areas. Starting in the 1930s, extreme value theory has identified universal properties of the statistics of extreme values of weakly dependent variables. For variables with strong correlations, the problem is more complex. However, significant progress has been made in the past twenty years. Like many breakthroughs in probability, new ideas have often emerged from a deep interplay with physics and statistical mechanics. The proposed research program continues in this spirit, aiming to identify and describe universality classes for the statistics of extreme values of highly correlated variables. The focus is on Gaussian processes in high dimensions and with strong correlations that arise in statistical mechanics. Important examples include spin glasses, such as the Sherrington-Kirkpatrick (SK) model and the Edwards-Anderson (EA) model, as well as branching Brownian motion (BBM) and the Gaussian free field (GFF). A body of bold conjectures was proposed in the 1980s by physicists to describe the complexity of the correlations of such systems. These questions remain largely open to this day. The long-term objectives aim to rigorously establish and extend the ideas of the theory. The directions for the next five years are: 1. Identify a new universality class for extreme values statistics, the class of log-correlated models, that would include BBM and GFF. 2. Find universal structures for the correlations between the variables that are close to the maximum at the level of the Gibbs measure, which concentrates on the extreme values. 3. For the Edwards-Anderson (EA) model on Z^d, describe the Gibbs measure of the process. In particular, establish the triviality of the model at d=2 (no phase transition) and the non-triviality of the model for higher d. Progress in these three directions will unveil core principles leading to an extension of the classical extreme value theory and of its applications.