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 Probability Surveys > Vol. 11 (2014) open journal systems 

Gaussian multiplicative chaos and applications: A review

Rémi Rhodes, University Paris-Dauphine
Vincent Vargas, University Paris-Dauphine

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.

AMS 2000 subject classifications: Primary 60G57; secondary 60G15, 28A80.

Keywords: Gaussian multiplicative chaos, review, KPZ, Gaussian process, multifractal measures.

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Rhodes, Rémi, Vargas, Vincent, Gaussian multiplicative chaos and applications: A review, Probability Surveys, 11, (2014), 315-392 (electronic). DOI: 10.1214/13-PS218.


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Probability Surveys. ISSN: 1549-5787