# Gaussian multiplicative chaos and applications: A review

@article{Rhodes2013GaussianMC, title={Gaussian multiplicative chaos and applications: A review}, author={R{\'e}mi Rhodes and Vincent Vargas}, journal={Probability Surveys}, year={2013}, volume={11}, pages={315-392} }

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…

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## References

SHOWING 1-10 OF 191 REFERENCES

Gaussian multiplicative chaos revisited

- Mathematics
- 2008

In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction…

Gaussian Multiplicative Chaos and KPZ Duality

- Mathematics
- 2013

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures…

Spectral Dimension of Liouville Quantum Gravity

- Mathematics, Physics
- 2014

This paper is concerned with computing the spectral dimension of (critical) 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove…

Maximum of a log-correlated Gaussian field

- Mathematics
- 2013

We study the maximum of a Gaussian field on $[0,1]^\d$ ($\d \geq 1$) whose correlations decay logarithmically with the distance. Kahane \cite{Kah85} introduced this model to construct mathematically…

Scaling exponents and multifractal dimensions for independent random cascades

- Mathematics, Physics
- 1996

This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(q), and (ii) multifractal dimensions are…

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

- Mathematics
- 2012

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching…

Levy multiplicative chaos and star scale invariant random measures

- Mathematics
- 2014

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of…

Values of Brownian intersection exponents, II: Plane exponents

- Mathematics
- 2001

This paper is the follow-up of the paper [27], in which we derived the exact value of intersection exponents between Brownian motions in a half-plane. In the present paper, we will derive the value…

Scale invariant random measures

- Mathematics
- 2012

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to char- acterize the law of…

On the heat kernel and the Dirichlet form of Liouville Brownian motion

- Mathematics
- 2013

In a previous work, a Feller process called Liouville Brownian motion on $\mathbb{R}^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the…