Gaussian multiplicative chaos and applications: a review

TL;DR

This review covers the development, extensions, and applications of Gaussian multiplicative chaos theory, highlighting recent advances like convergence of discretized measures and multifractal analysis across dimensions.

Contribution

It provides a comprehensive overview of Gaussian multiplicative chaos, including new results on discretized measures and multifractal analysis, connecting theory with diverse applications.

Findings

Convergence of discretized Liouville measures on isoradial graphs.

Multifractal analysis of measures in all dimensions.

Extensions of Gaussian multiplicative chaos theory to various fields.

Abstract

In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane's seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in 2d-Liouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from finance, through the Kolmogorov-Obukhov model of turbulence to 2d-Liouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.