Gaussian multiplicative chaos through the lens of the 2D Gaussian free field

TL;DR

This paper reviews the construction and properties of Gaussian multiplicative chaos measures derived from the 2D Gaussian free field, focusing on convergence, uniqueness, and scaling relations in the subcritical regime.

Contribution

It provides a unified, self-contained overview of GMC measures for 2D log-correlated fields, emphasizing the Gaussian free field and related scaling properties.

Findings

Convergence and uniqueness of GMC measures established

Kahane's convexity inequalities revisited

KPZ relation for scaling exponents presented

Abstract

The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.