The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

Ambjørn, J.; Budd, Timothy

doi:10.1016/j.physletb.2013.06.009

Cited by 7 publications

(12 citation statements)

References 28 publications

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“…The use of toroidal topology also allowed us to use loops with non-trivial winding numbers to extract information about the fractal structure of our spatial geometries. This has been done successfully in two-dimensional Euclidean quantum gravity [11], and also here in the four-dimensional case it provides us with a lot of information. In some sense, the shortest non-contractible loops can be considered as dual to the minimal cell boundaries and, like in the two-dimensional case, the shortest loops lie in narrow "valleys", where the surrounding "hills" look like (topologically) spherical outgrowths.…”

Section: Resultsmentioning

confidence: 93%

Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations

et al. 2019

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Causal Dynamical Triangulations is a non-perturbative quantum gravity model, defined with a lattice cut-off. The model can be viewed as defined with a proper time but with no reference to any three-dimensional spatial background geometry. It has four phases, depending on the parameters (the coupling constants) of the model. The particularly interesting behavior is observed in the so-called de Sitter phase, where the spatial three-volume distribution as a function of proper time has a semi-classical behavior which can be obtained from an effective minisuperspace action. In the case of the three-sphere spatial topology, it has been difficult to extend the effective semi-classical description in terms of proper time and spatial three-volume to include genuine spatial coordinates, partially because of the background independence inherent in the model. However, if the spatial topology is that of a three-torus, it is possible to define a number of new observables that might serve as spatial coordinates as well as new observables related to the winding numbers of the three-dimensional torus. The present paper outlines how to define the observables, and how they can be used in numerical simulations of the model.

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Section: Resultsmentioning

confidence: 93%

Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations

et al. 2019

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“…Matter correlation functions gave agreement with Watabiki's formula, but geometric measurements agreed better with d h = 4 for 0 < c < 1. Recently, simulations have been performed of DT on the torus coupled to the Ising model (c = 1/2) and the 3-states Potts model (c = 4/5) [18]. In addition to the shortest non-contractible loop length 0 , also the length 1 of the second shortest independent loop was analyzed (see Fig.…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…13. An example of two shortest, independent loops [18]. are expected, for large N , to be of the form…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

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Two-dimensional Quantum Geometry

Ambjørn¹,

Budd²

2013

Acta Phys. Pol. B

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“…Plots of k n pn{n 0 q 1{d W γ with the scaling parameters k n established via finite-size scaling of the graph distance (purple) and dual graph distance (green). The solid curves correspond to best fits of the ansatz(22).…”

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confidence: 99%

Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

Barkley¹,

Budd²

2019

Class. Quantum Grav.

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Two-dimensional quantum gravity, defined either via scaling limits of random discrete surfaces or via Liouville quantum gravity, is known to possess a geometry that is genuinely fractal with a Hausdorff dimension equal to 4. Coupling gravity to a statistical system at criticality changes the fractal properties of the geometry in a way that depends on the central charge of the critical system. Establishing the dependence of the Hausdorff dimension on this central charge c has been an important open problem in physics and mathematics in the past decades. All simulation data produced thus far has supported a formula put forward by Watabiki in the nineties. However, recent rigorous bounds on the Hausdorff dimension in Liouville quantum gravity show that Watabiki's formula cannot be correct when c approacheś 8. Based on simulations of discrete surfaces encoded by random planar maps and a numerical implementation of Liouville quantum gravity, we obtain new finite-size scaling estimates of the Hausdorff dimension that are in clear contradiction with Watabiki's formula for all simulated values of c P p´8, 0q. Instead, the most reliable data in the range c P r´12.5, 0q is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for c P p´8,´12.5q display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.

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The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

Cited by 7 publications

References 28 publications

Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations

Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations

Two-dimensional Quantum Geometry

Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

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