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V.: Gaussian multiplicative chaos and applications: a review, arxiv
"... 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 th ..."
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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 2dLiouville 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 KolmogorovObukhov model of turbulence to 2dLiouville 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.
V.: Lognormal ⋆scale invariant random measures, Probability Theory and Related Fields
"... Inthisarticle, weconsiderthecontinuousanalogofthecelebratedMandelbrotstarequation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. ..."
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Cited by 23 (7 self)
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Inthisarticle, weconsiderthecontinuousanalogofthecelebratedMandelbrotstarequation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects theconstraints imposed bythe continuoussetting. Inparticular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purpose, we define a Lévy multiplicative chaos that generalizes the already existing constructions. 1.
V.: Gaussian multiplicative chaos and KPZ duality
"... This paper is concerned with the KPZ formula. On the first hand, we give a simplified (in comparison with the existing literature) proof of the classical KPZ formula. On the other hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase ..."
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Cited by 19 (6 self)
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This paper is concerned with the KPZ formula. On the first hand, we give a simplified (in comparison with the existing literature) proof of the classical KPZ formula. On the other hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase (i.e. γ2> 2d). We prove the dual KPZ formula for these measures and check the duality relation. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity. 1.
PoissonDirichlet statistics for the extremes of a logcorrelated Gaussian field, preprint, arXiv:1203.4216 [math.PR
, 2012
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On the heat kernel and the Dirichlet form of Liouville Brownian Motion
"... In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field eγ X and is the right diffusion process to consider regarding 2dLiouville qua ..."
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Cited by 10 (2 self)
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In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field eγ X and is the right diffusion process to consider regarding 2dLiouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially [13] and the techniques introduced in [14]. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a nontrivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in [14] was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. In particular, the theory developed for example in [28, 29, 30], whose aim is to capture the “geometry ” of the underlying space out of the Dirichlet form of a process living on that space, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called intrinsic metric, which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.
V.: Glassy phase and freezing of logcorrelated Gaussian potentials, arXiv:1310.5574
"... In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the socalled glassy phase. The limiting Gibbs weights are integrated atomic r ..."
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Cited by 8 (2 self)
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In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the socalled glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos constructed in [10, 11]. This could be seen as a first rigorous step in the renormalization theory of supercritical Gaussian multiplicative chaos.
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 logcorrelated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the ..."
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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 logcorrelated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the socalled Tachyon fields, the conformally invariant operators in critical Liouville Quantum
Quantum gravity and the KPZ formula [after DuplantierSheffield], Sém. Bourbaki, 64e année 1052
, 2011
"... The study of statistical physics models in two dimensions (d = 2) at their critical point is in general a significantly hard problem (not to mention the d = 3 case). In the eighties, three physicists, Knizhnik, Polyakov and Zamolodchikov (KPZ) came up in [KPZ88] with a novel and farreaching approac ..."
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Cited by 2 (0 self)
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The study of statistical physics models in two dimensions (d = 2) at their critical point is in general a significantly hard problem (not to mention the d = 3 case). In the eighties, three physicists, Knizhnik, Polyakov and Zamolodchikov (KPZ) came up in [KPZ88] with a novel and farreaching approach in order to understand the critical behavior of