Probability Surveys

Gaussian multiplicative chaos and applications: A review

Rémi Rhodes and Vincent Vargas

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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.

Article information

Source
Probab. Surveys, Volume 11 (2014), 315-392.

Dates
First available in Project Euclid: 3 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.ps/1415023603

Digital Object Identifier
doi:10.1214/13-PS218

Mathematical Reviews number (MathSciNet)
MR3274356

Zentralblatt MATH identifier
1316.60073

Subjects
Primary: 60G57: Random measures
Secondary: 60G15, 28A80

Keywords
Gaussian multiplicative chaos review KPZ Gaussian process multifractal measures

Citation

Rhodes, Rémi; Vargas, Vincent. Gaussian multiplicative chaos and applications: A review. Probab. Surveys 11 (2014), 315--392. doi:10.1214/13-PS218. https://projecteuclid.org/euclid.ps/1415023603


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