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.

In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly formula. We also make precise conjectures about the relationship of the theory to scaling limits of random planar maps conformally embedded onto the sphere.

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 random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.1. Introduction. 1.1.Overview. In the 1980s, Kahane [45] developed a continuous parameter theory of multifractal random measures, called Gaussian multiplicative chaos; this theory emerged from the need to define rigorously the limit lognormal model introduced by Mandelbrot [59] in the context of turbulence. His efforts were followed by several authors [3,7,11,35,[67][68][69] [13] in the context of Mandelbrot's multiplicative cascades. This was done in the standard case of Liouville quantum gravity, namely strictly below the critical value of the GFF coupling constant γ in the Liouville conformal factor, that is, for γ < 2 (in a chosen normalization). Beyond this threshold, the standard construction yields vanishing random measures [29,45]. The issue of mathematically constructing singular Liouville measures beyond the phase transition (i.e., for γ > 2) and deriving the corresponding (nonstandard dual) KPZ formula has been investigated in [9,28,29], giving the first mathematical understanding of the so-called duality in Liouville quantum gravity; see [4,5,21,27,32,44,[48][49][50]54] for an account of physical motivations. However, the rigorous construction of random measures at criticality, that is, for γ = 2, does not seem to ever have been carried out.As stated above, once the Gaussian randomness is fixed, the standard Gaussian multiplicative chaos describes a random positive measure for each γ < 2 but yields 0 when γ = 2. Naively, one might therefore guess that −1 times the derivative at γ = 2 would be a random positive measure. This intuition leads one to consider the so-called derivative martingale, formally obtained by differentiating the standard measure w.r.t. γ at γ = 2, as explained below. In the case of branching Brownian motions [62], or of branching random walks [15,56] (see also [2] for a recent different but equivalent construction), the construction of such an object has already been carried out mathematically. In the context of branching random walks, the derivative martingale was introduced in the study of the fixed points of the smoothing transform at criticality (the smoothing transform is a generalization of Mandelbrot's ⋆-equation for discrete multiplicative cascades; see also [16]). Our construction will therefore appear as a continuous analogue of those works in the context of Gaussian multiplicative chaos.Besides the 2D-Liouville Quantum Gravity framework (and the KPZ formula), many other important models or qu...

Gaussian Multiplicative Chaos is a way to produce a measure on R d (or subdomain of R d ) of the form e γX(x) dx, where X is a log-correlated Gaussian field and γ ∈ [0, √ 2d) is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between −∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when γ = √ 2d. Two methods have been proposed to produce a non-trivial measure when γ = √ 2d. The first involves taking a derivative at γ = √ 2d (and was studied in an earlier paper by the current authors), while the second involves a modified renormalization scheme. We show here that the two constructions are equivalent and use this fact to deduce several quantitative properties of the random measure. In particular, we complete the study of the moments of the derivative martingale, which allows us to establish the KPZ

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\gamma X(z)}\,dz^2$, $\gamma<\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_t$ depending on the local behavior of the Liouville measure "$M_{\gamma}(dz)=e^{\gamma X(z)}\,dz$". We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for all $\gamma<\gamma_c$, the Liouville measure $M_{\gamma}$ is invariant under $P_{\mathbf{t}}$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.Comment: Published at http://dx.doi.org/10.1214/15-AOP1042 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org