Extrema of Log-correlated Random Variables: Principles and Examples

Abstract

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 Mathématiques 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 twodimensional 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.

The maximum deviation of the $\text{Sine} _\beta $ counting process

“…We are interested in the question of whether this function is an example of a process that should satisfy log-correlated field predictions. For an overview on work related to logcorrelated Gaussian and approximately Gaussian processes see [Arg16,Zei16]. This question follows naturally from the fact that the counting function of Sine β is a scaling limit of the imaginary part of the logarithm of the characteristic polynomial of random matrices.…”

The Fyodorov–Bouchaud formula and Liouville conformal field theory

“…A problem that has attracted a lot of attention is the behaviour of the maximum of the Gaussian free field on the unit circle. See for instance [3] for a review on extreme value statistics of logcorrelated processes. The link with GMC theory goes as follows, it is possible to make sense of GMC in the critical case γ = 2 by the so-called derivative martingale construction.…”

From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

“…Form this the proof of Theorem 1.1 follows in the case σ 1 < 1. 4 Note that this holds only if α > 0. As soon as 1 − σ 2 1 = O(1), the bridge condition disappears completely.…”

“…So-called log-correlated (Gaussian) processes have received considerable attention over the last years, see e.g. [27,4,2,8,9]. One of the reasons for this is that they represent processes where the correlations are on the borderline of becoming relevant for the properties of the extremes of the process.…”