Gaussian multiplicative chaos and applications: a review

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.

73 pages. Review with new results

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Cited by in corpus (15)

• Freezing and decorated Poisson point processes
• Liouville heat kernel: regularity and bounds
• On the heat kernel and the Dirichlet form of Liouville Brownian Motion
• A glimpse of the conformal structure of random planar maps
• Operator limits of random matrices
• Constructive Liouville Conformal Field Theory
• Liouville Quantum Gravity on the Riemann sphere
• Glassy phase and freezing of log-correlated Gaussian potentials
• Fractional Gaussian fields: a survey
• Modular bootstrap agrees with path integral in the large moduli limit
• Dual Scale Invariance Spontaneous Symmetry Breaking and Turbulence
• Liouville Brownian motion at criticality
• Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian Free Field
• Quadratic Fluctuations of the Simple Exclusion Process
• Thick Points of High-Dimensional Gaussian Free Fields