Maxima of log-correlated fields: some recent developments* *We dedicate this review to the memory of Fritz Haake, who did so much to understand and explain the connections between Random Matrix Theory and Quantum Chaos, and whose friendship one of us (JPK) was greatly fortunate to enjoy.

Abstract

We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.