Gaussian multiplicative chaos and applications: a review
type: Item
instance of: scholarly article
title: Gaussian multiplicative chaos and applications: a review
zbMATH Open document ID: 1316.60073
DOI: 10.1214/13-PS218
author: Rémi Rhodes
author: Vincent Vargas
published in: Probability Surveys
publication date: 2014-11-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.6221
full work available at URL: https://projecteuclid.org/euclid.ps/1415023603
review text: This rich and interesting survey begins with the seminal work by \textit{J.-P. Kahane} [Ann. Sci. Math. Qué. 9, 105--150 (1985; Zbl 0596.60041)], which rigorously introduced the standard Gaussian multiplicative chaos, i.e., the random measure \(M_\gamma\) on \(D \subset\mathbb R^d\) defined (at least formally) by: \[ M_\gamma (A) :=\int_A e^{\gamma X(x)-\frac12 \gamma^2\mathbb E[X(x)^2]}\sigma dx, \] where \(\sigma\) is a fixed measure on \(D\), and \(X = (X(x), x \in D)\) is a centred Gaussian field, with correlation \(\mathbb E[X(x)X(y)] \sim \log^+ (\frac{1}{|x-y|})\) as \(|x-y|\to 0\); moreover, \(\gamma < \gamma_c = \sqrt{2d}\) if \(\sigma\) is the Lebesgue measure. Kahane defined \(M_\gamma\) through a limiting procedure, considering an always slighter cut-off of the field \(X\). Then other regularization procedures have been considered, mainly the convolution of \(X\) by a kernel going to the Dirac mass \(\delta_0\). The following related questions are also discussed in this survey: {\parindent=6mm \begin{itemize}\item[--] Does the limiting measure \(M_\gamma\) depend on the chosen cut-off procedure? \item[--] What are the geometrical and statistical properties of the measure \(M_\gamma\)? \item[--] What are the regularity properties (multifractal analysis) of the measure \(M_\gamma\)? \item[--] How can we characterize the measure \(M_\gamma\)? \item[--] What happens at \(\gamma = \gamma_c\)? And what about \(\gamma > \gamma_c\)? \end{itemize}} The case of \(X\) being a Gaussian free field has also been extensively analyzed, and is considered in detail in this survey. A central application of this important case concerns the \(2d\)-Liouville quantum gravity, an attempt to construct a canonical ``Riemannian'' random metric on the sphere (related to the so-called Brownian map). As a rigorous construction of this random metric seems to lay beyond the actual possibilities of the theory, instead of the metric itself, the Gaussian multiplicative chaos theory allows…
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60-02
zbMATH DE Number: 6370363
zbMATH Keywords: Gaussian multiplicative chaos
zbMATH Keywords: Gaussian free field
zbMATH Keywords: random measures
zbMATH Keywords: multifractal measures
zbMATH Keywords: 2d-Liouville quantum gravity
zbMATH Keywords: random metric
zbMATH Keywords: random Liouville measure
zbMATH Keywords: geometrical KPZ formulae
MaRDI profile type: Publication
arXiv ID: 1305.6221
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Gaussian multiplicative chaos and applications: a review (OpenAlex ID)
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