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

## TL;DR

This paper introduces new methods to construct Gaussian multiplicative chaos measures for the 2D Gaussian free field, linking multiplicative cascades with GMC and confirming conjectures about CLE$_4$-based measures.

## Contribution

It provides novel constructions of subcritical and critical GMC measures using local sets of the GFF, establishing a connection with multiplicative cascades and confirming a conjecture relating CLE$_4$ measures to GMC.

## Key findings

- Constructed GMC measures via local sets of GFF.
- Linked multiplicative cascades with GMC measures.
- Confirmed CLE$_4$ measures are law-equivalent to GMC.

## 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.