Liouville quantum gravity on the annulus

Rémy, Guillaume

doi:10.1063/1.5030409

Cited by 17 publications

(17 citation statements)

References 30 publications

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“…Each of these measures will naturally be called from then on Liouville measures. See the works [HRV15, CRV16,Rem17] for extensions of this construction to other topologies (disk, genus g ≥ 2, annuli) and the works [DK+15, KRV15,Dozz] for striking recent results on the structure and properties of the Liouville field (DOZZ formula etc). In the case of S 2 , a different viewpoint on the Liouville field is provided by the works [She16,DMS14].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Liouville

Garban

2020

Journal of Functional Analysis

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The aim of this paper is to analyze an SPDE which arises naturally in the context of Liouville quantum gravity. This SPDE shares some common features with the so-called Sine-Gordon equation and is built to preserve the Liouville measure which has been constructed recently on the two-dimensional sphere S 2 and the torus T 2 in the work by David-Kupiainen-Rhodes-Vargas [DK+16, DRV16]. The SPDE we shall focus on has the following (simplified) form:The main aspect which distinguishes this singular stochastic SPDE with well-known SPDEs studied recently (KPZ, dynamical Φ 4 3 , dynamical Sine-Gordon, generalized KPZ, etc.) is the presence of intermittency. One way of picturing this effect is that a naive rescaling argument suggests the above SPDE is sub-critical for all γ > 0, while we don't expect solutions to exist when γ > 2. In this work, we initiate the study of this intermittent SPDE by analyzing what one might call the "classical" or "Da Prato-Debussche" phase which corresponds here to γ ∈ [0, γ dP D = 2 √ 2 − √ 6). By exploiting the positivity of the non-linearity e γX , we can push this classical threshold further and obtain this way a weaker notion of solution when γ ∈ [γ dP D , γ pos = 2 √ 2 − 2). Our proof requires an analysis of the Besov regularity of natural space/time Gaussian multiplicative chaos (GMC) measures. Regularity Structures of arbitrary high degree should potentially give strong solutions all the way to the same threshold γ pos and should not push this threshold further unless the notion of regularity is suitably adapted to the present intermittent situation. Of independent interest, we prove along the way (using techniques from [HS16]) a stronger convergence result for approximate GMC measures µ → µ which holds in Besov spaces.

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Section: Introductionmentioning

confidence: 99%

Dynamical Liouville

Garban

2020

Journal of Functional Analysis

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“…In [11] the authors discovered that the correlation functions of LCFT could be expressed as fractional moments of GMC measures with log singularities therefore rendering possible the mathematical study of LCFT. The theory was defined on the Riemann sphere in [11] then on the unit disk in [19] and on other surfaces in [10], [29]. Let us also mention another interesting approach by Duplantier-Miller-Sheffield [13] which develops a theory of quantum surfaces with two marked points linked to the two-point correlation function of LCFT (see [34] for a precise statement of this connection).…”

Section: Strategy Of the Proofmentioning

confidence: 99%

The Fyodorov–Bouchaud formula and Liouville conformal field theory

Rémy¹

2020

Duke Math. J.

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In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle [17]. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk defined by Huang, Rhodes and Vargas [19]. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (BPZ equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF and tail expansions of GMC.

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“…Consider an annulus represented as a cylinder of length πτ and of radius 1. Let q = e −2πτ and Z Annulus be the partition function (no insertion points) of LCFT on this annulus defined probabilistically in an analogous way to (1.7), see also [Rem18]. Then the following equality should hold:…”

Section: Perspectives and Outlookmentioning

confidence: 99%

“…The fundamental building block of his framework is the Liouville conformal field theory (LCFT), which describes the law of the conformal factor of the metric tensor of a surface of fixed complex structure. LCFT was first made rigorous in probability theory in the case of the Riemann sphere in [DKRV16], and then in the case of a simply connected domain with boundary in [HRV18] (see also [DRV16,Rem18,GRV19] for the case of other topologies).…”

Section: Introductionmentioning

confidence: 99%

FZZ formula of boundary Liouville CFT via conformal welding

Ang¹,

Rémy²,

Sun³

2021

Preprint

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Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov (2000) proposed an explicit expression, the so-called the FZZ formula, for the one-point bulk structure constant for LCFT on the disk. In this paper we give a proof of the FZZ formula in the probabilistic framework of LCFT, which represents the first step towards rigorously solving boundary LCFT using conformal bootstrap. In contrast to previous works, our proof is based on conformal welding of quantum disks and the mating-of-trees theory for Liouville quantum gravity. As a byproduct of our proof, we also obtain the exact value of the variance for the Brownian motion in the mating-of-trees theory. Our paper is an essential part of an ongoing program proving integrability results for Schramm-Loewner evolutions, LCFT, and in the mating-of-trees theory.

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Liouville quantum gravity on the annulus

Cited by 17 publications

References 30 publications

Dynamical Liouville

Dynamical Liouville

The Fyodorov–Bouchaud formula and Liouville conformal field theory

FZZ formula of boundary Liouville CFT via conformal welding

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