We consider the discrete Gaussian Free Field in a square box in ( Z ^2 ) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as ( N ). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever ( r_N ) and ( r_N N 0 ), to a Poisson point process with intensity measure ( Z ( dx) e ^ - h d h ), where ( := 2 g ) with g: = 2 π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.
@cite_30: 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 logcorrelated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.
@cite_18: Gaussian Multiplicative Chaos is a way to produce a measure on ( R ^d ) (or subdomain of ( R ^d )) of the form ( e^ X(x) dx ), where X is a log-correlated Gaussian field and ( [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 ( = 2d ).
@cite_22: 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 is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent flow.
@cite_48: 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 nance, 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.
@cite_0: Chaos additif et chaos multiplicatif. Theoremes de comparaison et d'unicite. Exemples et applications. Les theories L 2 et L 2m . Application de la theorie L 2 a la non-degenerescence de Q et aux proprietes de S=Qσ. La probabilite de Peyriere. Cas de degenerescence
@cite_12: In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.
A version of the Gibbs measure appears also in the studies of Gaussian multiplicative chaos by Robert and Vargas @cite_22 , Allez, Rhodes and Vargas @cite_12 and Duplantier, Rhodes, Sheffield and Vargas @cite_30 @cite_18 ; see Rhodes and Vargas @cite_48 for a recent review of this subject going back to Kahane @cite_0 . In particular, the papers @cite_30 @cite_18 are concerned with the construction of a which corresponds to the @math limit of (unnormalized) measure This is an object closely related to the @math -derivative of @math at @math . On the basis of various conjectural statements, the references @cite_30 @cite_18 predict the derivative martingale to appear in the place of our @math . In Theorem we thus prove a version of this prediction for the DGFF. (The apparent discrepancy in the factor @math in front of the variance in the exponents of and comes from the fact that the measure @math in has been scaled by a factor of @math .)