We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon’s construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques used for multiplicative cascades to the study of GMC measures of the GFF. As a proof of principle we do this for the so-called Seneta–Heyde rescaling of the critical GMC measure.
On propose de nouvelles constructions des mesures du chaos multiplicatif gaussien (GMC) sous-critique et critique correspondant au champ libre gaussien 2D (GFF). Comme cas particulier, on retrouve la construction des mesures aléatoires par E. Aidekon, qui utilise des ensembles de boucles emboîtées invariantes par transformations conformes. Ainsi, on prouve sa conjecture selon laquelle certaines mesures basées sur le CLE emboîté sont égal en loi aux mesures de GMC pour le GFF. Nos constructions sont basées sur la théorie des ensembles locaux du GFF et permettent d’établir un lien fort entre les cascades multiplicatives et les mesures GMC. Ce lien nous permet d’adapter directement les techniques utilisées pour les cascades multiplicatives à l’étude des mesures de GMC pour le GFF. Comme exemple de ce principe on adapte l’argument de Seneta–Heyde pour construire la mesure critique de la GMC.
Revised : 2017-12-27
Accepted : 2019-01-17
Classification: 60G57, 60G58, 60D05, 60J80
Keywords: Liouville measure, Gaussian free field, Gaussian multiplicative chaos, critical Gaussian multiplicative chaos, multiplicative cascades
@unpublished{AIF_0__0_0_A5_0,
author = {Aru, Juhan and Powell, Ellen and Sep\'ulveda, Avelio},
title = {Liouville measure as a multiplicative cascade via level sets of the Gaussian free field},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Aru, Juhan; Powell, Ellen; Sepúlveda, Avelio. Liouville measure as a multiplicative cascade via level sets of the Gaussian free field. Annales de l'Institut Fourier, to appear, 41 p.
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