Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

Barkley, J.; Budd, Timothy

doi:10.1088/1361-6382/ab4f21

“…(1.17) Recent numerical simulations [BB19] fit much better with (1.17) than with (1.16); and unlike (1.16) the formula (1.17) is consistent with all rigorously known bounds. However, there is currently no theoretical justification for (1.17) (even at a heuristic level).…”

Section: Resultsmentioning

confidence: 56%

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

Gwynne

¹

,

Holden

²

,

Pfeffer

³

et al. 2020

Commun. Math. Phys.

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There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge c M ∈ (−∞, 1]. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating γ times a variant of the planar Gaussian free field, where γ ∈ (0, 2] satisfies c M = 25 − 6(2/γ + γ/2) 2 . Physics considerations suggest that LQG should also make sense in the regime when c M > 1. However, the behavior in this regime is rather mysterious in part because the corresponding value of γ is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting.We introduce and study a discretization of LQG which makes sense for all values of c M ∈ (−∞, 25). Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same "LQG size" with respect to the Gaussian free field. We prove that several formulas for dimension-related quantities are still valid for c M ∈ (1, 25), with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for c M ∈ (1, 25), which gives a finite quantum dimension if and only if the Euclidean dimension is at most (25 − c M )/12. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius r grows faster than any power of r (which suggests that the Hausdorff dimension of LQG in the case when c M ∈ (1, 25) is infinite).We include a substantial list of open problems.

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“…The formula (1.2) is consistent with all known bounds for d γ . Moreover, recent numerical simulations by Barkley and Budd [BB19] fit much more closely with (1.2) than with (1.1). However, there is currently no theoretical justification, even at a heuristic level, for (1.2).…”

Section: Introduction 1overviewmentioning

confidence: 59%

The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

Gwynne

¹

2020

Commun. Math. Phys.

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Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

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“…As one of its most intricate features, the composite operator formalism employed in this work could act as a connector between Asymptotic Safety [22,23] and more geometric approaches to quantum gravity based on causal dynamical triangulations [80,81] or random geometry. In d = 2 dimensions, a natural benchmark would involve a quantitative comparison of scaling properties associated with the geodesic length recently considered in Pagani and Reuter [19], Becker and Pagani [29,30], Becker et al [31], and Houthoff et al [32] and exact computations for random discrete surfaces in the absence of matter fields [21,82] as well as rigorous and numerical bounds arising from Liouville Gravity in the presence of matter [83,84]. On the renormalization group side this will involve taking limits akin to Nink and Reuter [85].…”

Section: Discussionmentioning

confidence: 99%

On Characterizing the Quantum Geometry Underlying Asymptotic Safety

Kurov

¹

,

Saueressig

²

2020

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The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a "perturbative regime" where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R 2 -operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.

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Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

Cited by 8 publications

References 59 publications

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

On Characterizing the Quantum Geometry Underlying Asymptotic Safety

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