Madaule, Thomas
Maximum of a log-correlated gaussian field
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4 , p. 1369-1431
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consulter l'article sur le site de la revue
MR 3414451
doi : 10.1214/14-AIHP633
URL stable : http://www.numdam.org/item?id=AIHPB_2015__51_4_1369_0

Nous étudions le maximum d’un champ Gaussien sur [0,1] 𝚍 (𝚍1) 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).
We study the maximum of a Gaussian field on [0,1] 𝚍 (𝚍1) 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.

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