Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

Juhan Aru (1) , Ellen Powell (1) , Avelio Sepúlveda (2)

1 ETH Zürich - Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich]
2 PSPM - Probabilités, statistique, physique mathématique

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$_4$ 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.

Domaines: Probabilités [math.PR]

Juhan Aru, Ellen Powell, Avelio Sepúlveda. Liouville measure as a multiplicative cascade via level sets of the Gaussian free field. Annales de l'Institut Fourier, 2020, 70 (1), pp.205-245. ⟨10.5802/aif.3312⟩. ⟨hal-02080127⟩