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26
2013): Extreme local extrema of twodimensional discrete Gaussian free field. arXiv preprint arXiv:1306.2602
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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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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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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
Exact dimensionality and projections of random selfsimilar measures and sets
 J. London Math. Soc
, 2014
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Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble
"... The goal of this paper is to establish a relation between characteristic polynomials of N ×N GUE random matrices H as N → ∞, and Gaussian processes with logarithmic correlations. First, we introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian proces ..."
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The goal of this paper is to establish a relation between characteristic polynomials of N ×N GUE random matrices H as N → ∞, and Gaussian processes with logarithmic correlations. First, we introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN (z): = − log det(zI−H)  on mesoscopic scales as N →∞. By employing a Fourier integral representation, we show how this implies a continuous analogue of a result by Diaconis and Shahshahani [18]. On the macroscopic scale, DN (x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give
Inverse problems in multifractal analysis
, 2013
"... Abstract. Multifractal formalism is designed to describe the distribution at small scales of the elements of M+c (Rd), the set of positive, finite and compactly supported Borel measures on Rd. It is valid for such a measure µ when its Hausdorff spectrum is the upper semicontinuous function given by ..."
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Abstract. Multifractal formalism is designed to describe the distribution at small scales of the elements of M+c (Rd), the set of positive, finite and compactly supported Borel measures on Rd. It is valid for such a measure µ when its Hausdorff spectrum is the upper semicontinuous function given by the concave LegendreFenchel transform of the free energy function τµ associated with µ; this is the case for fundamental classes of exact dimensional measures. For any function τ candidate to be the free energy function of some µ ∈M+c (Rd), we build such a measure, exact dimensional, and obeying the multifractal formalism. This result is extended to a refined formalism considering jointly Hausdorff and packing spectra. Also, for any upper semicontinuous function candidate to be the lower Hausdorff spectrum of some exact dimensional µ ∈M+c (Rd), we build such a measure. Our results transfer to the analoguous inverse problems in multifractal analysis of Hölder continuous functions. Contents
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.