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arXiv · 1701.05872

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

Juhan Aru·Ellen Powell·Avelio Sepúlveda
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Abstract

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.

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Juhan Aru, Ellen Powell, Avelio Sepúlveda. 2017-12-27. Liouville measure as a multiplicative cascade via level sets of the Gaussian free field. https://doi.org/10.5802/aif.3312

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