Extrema of log-correlated random variables: Principles and Examples

TL;DR

This paper reviews recent advances in understanding the extreme values of log-correlated random fields, highlighting examples like branching random walks and the Gaussian free field, and discusses their implications for number theory and random matrix theory.

Contribution

It surveys the properties of log-correlated fields and introduces a multiscale second moment method to accurately determine their maximum values.

Findings

Established the leading and subleading order of maxima for branching random walks.

Connected the extremal behavior of these fields to conjectures in number theory and random matrix theory.

Provided a unified approach to analyze the extremes of log-correlated fields.

Abstract

These notes were written for the mini-course "Extrema of log-correlated random variables: Principles and Examples" at the Introductory School held in January 2015 at the Centre International de Rencontres Math\'ematiques in Marseille. There have been many advances in the understanding of the high values of log-correlated random fields from the physics and mathematics perspectives in recent years. These fields admit correlations that decay approximately like the logarithm of the inverse of the distance between index points. Examples include branching random walks and the two-dimensional Gaussian free field. In this paper, we review the properties of such fields and survey the progress in describing the statistics of their extremes. The branching random walk is used as a guiding example to prove the correct leading and subleading order of the maximum following the multiscale refinement of the second moment method of Kistler. The approach sheds light on a conjecture of Fyodorov, Hiary & Keating on the maximum of the Riemann zeta function on an interval of the critical line and of the characteristic polynomial of random unitary matrices.