Gaussian multiplicative chaos and applications: a review
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 to rigorously construct the associated volume form, namely, the so-called (random) Liouville measure. Thence, on the one hand, important geometrical KPZ formulae are derived, and on the other hand, the whole procedure is generalized to other types of Gaussian fields \(X\). The authors carefully expose all this matter, and moreover, they show that the Liouville measure can be approached by discrete versions of it on triangular graphs. Finally, they investigate the case \(\gamma^2\geq \gamma_c^2 = 2d\), too, leading to ``critical Gaussian multiplicative chaos'' and to ``atomic Gaussian multiplicative chaos'', and then in particular to their relationships to duality in \(2d\)-Liouville quantum gravity.
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