The Hausdorff Dimension of Two-Dimensional Quantum Gravity

Duplantier, Bertrand

doi:10.48550/arxiv.1108.3327

Cited by 4 publications

(6 citation statements)

References 26 publications

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“…An entry • indicates that the exponent is unknown. At the time of writing, there is no consensus about the value of the Hausdorff dimension d H in the physics literature, although a so called Watabiki formula has been proposed (see e.g., [24,41,2,3] and references therein) and critically analysed in view of recent mathematical results [37,70]. All other exponents can be derived rigorously in the O(n) model on triangulations, as well as the model with bending energy, and are expected to be universal.…”

Section: Phase Diagram and Critical Pointsmentioning

confidence: 99%

“…While this last question seems at present to be out of reach, its answer is expected to be related to the value of the almost sure Hausdorff dimension of large random maps with an O(n) model, a question which is under active debate (see, e.g., Refs. [2,3,37,41,70]).…”

Section: Introductionmentioning

confidence: 99%

“…Ref. [41] relates it to the value of yet another critical exponent, which expresses how deep geodesics enter in the nested configuration of loops.…”

mentioning

confidence: 99%

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Nesting statistics in the $O(n)$ loop model on random planar maps

Borot¹,

Bouttier²,

Duplantier³

2016

Preprint

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In the O(n) loop model on random planar maps, we study the depthin terms of the number of levels of nesting -of the loop configuration, by means of analytic combinatorics. We focus on the 'refined' generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O(n) loop model, we show that these generating series satisfy functional relations obtained by a modification of those satisfied by the unrefined generating series. In a more specific O(n) model where loops cross only triangles and have a bending energy, we explicitly compute the refined generating series. We analyze their non generic critical behavior in the dense and dilute phases, and obtain the large deviations function of the nesting distribution, which is expected to be universal. Using the framework of Liouville quantum gravity (LQG), we show that a rigorous functional KPZ relation can be applied to the multifractal spectrum of extreme nesting in the conformal loop ensemble (CLE κ ) in the Euclidean unit disk, as obtained by Miller, Watson and Wilson [111], or to its natural generalization to the Riemann sphere. It allows us to recover the large deviations results obtained for the critical O(n) random planar map models. This offers, at the refined level of large deviations theory, a rigorous check of the fundamental fact that the universal scaling limits of random planar map models as weighted by partition functions of critical statistical models are given by LQG random surfaces decorated by independent CLEs.

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Section: Phase Diagram and Critical Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nesting statistics in the $O(n)$ loop model on random planar maps

Borot¹,

Bouttier²,

Duplantier³

2016

Preprint

Self Cite

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“…Finite size analysis using the spin correlators seemed to favor d W h (c) although with somewhat large error bars, while the use of geometric quantity favored the d h = 4 hypothesis for c ≥ 0. Recently arguments have been given [15] in favor of d h = 4.…”

Section: Introductionmentioning

confidence: 99%

The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

Ambjørn

Budd

2013

Physics Letters B

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“…From this framework and the tools developed within, a wealth of implications on integrability and statistical mechanics followed and then led to the renown of matrix models [1]. The framework of random matrices still attracts a lot of attention in both physicist and mathematician communities [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Renormalizable Models in Rank $${d \geq 2}$$ d ≥ 2 Tensorial Group Field Theory

Geloun

2014

Commun. Math. Phys.

138

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Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank d tensor fields are defined over d copies of a group manifold GD = U (1) D or GD = SU (2) D with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form p 2a , a ∈]0, 1]. This study is tailored for models equipped with Laplacian dynamics on GD (case a = 1) but also for more exotic nonlocal models in quantum topology (case 0 < a < 1). A generic model can be written ( dim G D Φ k d , a), where k is the maximal valence of its interactions. Using a multi-scale analysis for the generic situation, we identify several classes of renormalizable actions including matrix model actions. In this specific instance, we find a tower of renormalizable matrix models parametrized by k ≥ 4. In a second part of this work, we focus on the UV behavior of the models up to maximal valence of interaction k = 6. All rank d ≥ 3 tensor models proved renormalizable are asymptotically free in the UV. All matrix models with k = 4 have a vanishing β-function at one-loop and, very likely, reproduce the same feature of the Grosse-Wulkenhaar model [Commun. Math. Phys. 256, 305 (2004)].

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The Hausdorff Dimension of Two-Dimensional Quantum Gravity

Cited by 4 publications

References 26 publications

Nesting statistics in the $O(n)$ loop model on random planar maps

Nesting statistics in the $O(n)$ loop model on random planar maps

The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

Renormalizable Models in Rank $${d \geq 2}$$ d ≥ 2 Tensorial Group Field Theory

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