Maximum of a log-correlated Gaussian field

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.