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… 
View PDF on arXiv
332 Citations
Highly Influential Citations
32
Background Citations
137
Methods Citations
43
Results Citations
7

Figures from this paper

332 Citations

Citation Type

Citation Type

Gaussian multiplicative chaos through the lens of the 2D Gaussian free field
The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for…
Gaussian multiplicative chaos and applications to Liouville quantum gravity
In this thesis, we study the theory of Liouville Quantum Gravity via probabilist approach, introduced in the seminal paper of Polyakov in 1981, using path integral formalism on 2d surfaces. To define…
The distribution of Gaussian multiplicative chaos on the unit interval
We consider the sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure.…
Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model
In this article we study imaginary Gaussian multiplicative chaos -- namely a family of random generalized functions which can formally be written as $e^{i X(x)}$, where $X$ is a log-correlated…
Complex Gaussian Multiplicative Chaos
In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all…
Universal tail profile of Gaussian multiplicative chaos
In this article we study the tail probability of the mass of Gaussian multiplicative chaos. With the novel use of a Tauberian argument and Goldie’s implicit renewal theorem, we provide a unified…
Scaling limits of critical systems in random geometry
This thesis focusses on the properties of, and relationships between, several fundamental objects arising from critical physical models. In particular, we consider Schramm–Loewner evolutions, the…
Tail universality of critical Gaussian multiplicative chaos
In this article we study the tail probability of the mass of critical Gaussian multiplicative chaos (GMC) associated to a general class of log-correlated Gaussian fields in any dimension, including…
Semiclassical limit of Liouville Field Theory
Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity
The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 191 REFERENCES
Gaussian multiplicative chaos revisited
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
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
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
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
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
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
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
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
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
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…
...
1
2
3
4
5
...