Maxima of log-correlated fields: some recent developments

TL;DR

This paper reviews recent advances in understanding the extreme value behavior of log-correlated fields, including random matrix characteristic polynomials and the Riemann zeta-function, highlighting conjectures, moments, and connections to Gaussian multiplicative chaos.

Contribution

It synthesizes recent developments linking extreme value statistics of various log-correlated fields with conjectures and theories like Gaussian multiplicative chaos and symmetric functions.

Findings

Progress on conjectures of Fyodorov & Keating

Connections established between extreme values and Gaussian multiplicative chaos

Explicit formulae derived from symmetric function theory

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 \& Keating concerning the extreme value statistics, moments of moments, connections to Gaussian Multiplicative Chaos, and explicit formulae derived from the theory of symmetric functions.