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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.
2013): Extreme local extrema of twodimensional discrete Gaussian free field. arXiv preprint arXiv:1306.2602
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Extreme values for twodimensional discrete Gaussian free field
, 2012
"... We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and precise (in the ..."
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We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and precise (in the exponential scale) estimate for the right tail on the law of the centered maximum. 1
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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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.
Maximum of a logcorrelated Gaussian field
"... Abstract. We study the maximum of a Gaussian field on [0, 1]d (d ≥ 1) whose correlations decay logarithmically with the distance. Kahane [22] introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas [19] [2 ..."
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Abstract. We study the maximum of a Gaussian field on [0, 1]d (d ≥ 1) whose correlations decay logarithmically with the distance. Kahane [22] introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas [19] [20] extended Kahane’s construction to the critical case and established the KPZ formula at criticality. Moreover, they made in [19] several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in [19]: we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. 1
Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise
"... Abstract Motivated by the general problem of studying sampletosample fluctuations in disordergenerated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model—the ideal periodic 1/f Gaussian noise. Our main object o ..."
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Abstract Motivated by the general problem of studying sampletosample fluctuations in disordergenerated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model—the ideal periodic 1/f Gaussian noise. Our main object of interest is the number of points NM(x) above a level x2 Vm, with Vm = 2 lnM standing for the leadingorder typical value of the absolute maximum for the sample of M points. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of NM(x) for 0 < x < 2. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level x, results in an important difference between the mean and the typical values of NM(x). This can be further used to determine the typical threshold xm of extreme values in the pattern which turns out to be given by x(typ)m = 2 − c ln lnM / lnM with c = 32. Such observation provides a rather compelling explanation of the mechanism behind universality of c. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disordergenerated multifractal fields. In particular, we predict that the typical value of the maximum pmax of intensity is to be given by
The glassy phase of complex branching Brownian motion
, 2013
"... In this paper, we study complex valued branching Brownian motion in the socalled glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on com ..."
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In this paper, we study complex valued branching Brownian motion in the socalled glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so called derivative martingale, mainly due to the clustering of the modulus. The proof relies on the fine description of the extremal process available in the branching Brownian
CONFORMAL SYMMETRIES IN THE EXTREMAL PROCESS OF TWODIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
"... Abstract: We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and show how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions (x) and centered values (h) of the extreme ..."
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Abstract: We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and show how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions (x) and centered values (h) of the extreme local maxima in lattice versions of a bounded domain D⊂C converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure ZD(dx) ⊗ e−αhdh, where α is a constant derived from overall normalization of the field and ZD is a random a.s.finite measure on D. The laws of the measures {ZD} are naturally interrelated; restrictions to subdomains are governed by a GibbsMarkov property and images under analytic bijections f by the transformation rule (Z f (D) ◦ f)(dx) law =  f ′(x)4ZD(dx). Conditions are also given that determine the laws of these measures uniquely. All but one of these are known to hold for the critical Liouville Quantum Gravity measure associated with the Continuum Gaussian Free Field. 1.
POISSONDIRICHLET STATISTICS FOR THE EXTREMES OF THE TWODIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
, 2013
"... Abstract. In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a logcorrelated Gaussian field converge to a PoissonDirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on sh ..."
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Abstract. In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a logcorrelated Gaussian field converge to a PoissonDirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a onestep replica symmetry breaking in spin glass terminology. This implies PoissonDirichlet statistics by general spin glass arguments. In this note, this approach is used to prove PoissonDirichlet statistics for the twodimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis. 1.
Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale
, 2012
"... In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the socalled 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 als ..."
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In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the socalled 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 logcorrelated Gaussian random variables. 1.