Maximum of a log-correlated Gaussian field

TL;DR

This paper proves the convergence in law of the maximum of a log-correlated Gaussian field, showing it follows a Gumbel distribution convolved with the derivative martingale limit, resolving a key conjecture in the field.

Contribution

It establishes the convergence in law of the maximum of a log-correlated Gaussian field and identifies its limiting distribution, resolving a major conjecture.

Findings

Maximum converges in law to a Gumbel distribution convolved with the derivative martingale limit.

Resolved a key conjecture on the maximum of log-correlated Gaussian fields.

Extended understanding of Gaussian multiplicative chaos at criticality.

Abstract

We study the maximum of a Gaussian field on $[0,1]^\d$ ($\d \geq 1$) whose correlations decay logarithmically with the distance. Kahane \cite{Kah85} introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas \cite{DRSV12a} \cite{DRSV12b} extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in \cite{DRSV12a} several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in \cite{DRSV12a}: we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale.