Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Maximum of a log-correlated Gaussian field

Thomas Madaule

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Abstract

We study the maximum of a Gaussian field on $[0,1]^{\mathtt{d}}$ ($\mathtt{d}\geq1$) whose correlations decay logarithmically with the distance. Kahane (Ann. Sci. Math. Québec 9 (1985) 105–150) introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) extended Kahane’s construction to the critical case and established the KPZ formula at criticality. Moreover, they made in (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint) several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint): 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.

Résumé

Nous étudions le maximum d’un champ Gaussien sur $[0,1]^{\mathtt{d}}$ ($\mathtt{d}\geq1$) dont les corrélations décroissent logarithmiquement avec la distance. Kahane (Ann. Sci. Math. Québec 9 (1985) 105–150) a introduit ce modèle pour construire mathématiquement le chaos Gaussien multiplicatif dans le cas sous-critique. Duplantier, Rhodes, Sheffield et Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) ont étendu cette construction au cas critique et ont établi la formule KPZ. De plus, dans (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint), ils fournissent plusieurs conjectures sur le cas sur-critique ainsi que sur le maximum de ce champ Gaussien. Dans ce papier nous établissons la convergence en loi du maximum et montrons que loi limite est une variable aléatoire de Gumbel convoluée avec la limite de la martingale dérivée, résolvant ainsi la Conjecture 12 de (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 4 (2015), 1369-1431.

Dates
Received: 23 August 2013
Revised: 28 April 2014
Accepted: 3 July 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432046

Digital Object Identifier
doi:10.1214/14-AIHP633

Mathematical Reviews number (MathSciNet)
MR3414451

Zentralblatt MATH identifier
1329.60138

Subjects
Primary: 60G15: Gaussian processes 60F05: Central limit and other weak theorems

Keywords
Gaussian multiplicative chaos Log-correlated Gaussian field Minimal position Gumbel distribution

Citation

Madaule, Thomas. Maximum of a log-correlated Gaussian field. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1369--1431. doi:10.1214/14-AIHP633. https://projecteuclid.org/euclid.aihp/1445432046


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