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

Understand

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.