"We propose to continue the study of the extremal structure of logarithmically correlated random fields, such as the (discrete or continuous) Gaussian free field and the positions of particles in a branching random walk or branching Brownian motion. Such study includes finding the complete asymptotics of the extremal process of the 2D discrete GFF (based on work started in Biskup-Louidor (2013)), studying properties of the limit and extending these results to other log-correlated fields. This will the first time extremal asymptotics are derived for log-correlated fields. Based on these, we then propose to study exponentials of log-correlated fields, such as the ones appearing in the Gaussian multiplicative chaos, the Liouville quantum gravity measure and the logarithmic random energy model, at least at sub-critical temperatures, thereby resolving several conjectures about their existence, asymptotics and properties. Finally, using the above, we wish to study the Glauber dynamics for the log random energy model, or more generally the motion of a particle in a log-correlated potential, in and out-of equilibrium and at least at sub-critical temperatures. Results as in the (non-logarithmic) random energy model, such as converges to a K-process and the fractional kinetics process are expected at first order, necessitating more subtle observables."
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