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 $c$ approaches $-\infty$. 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 $c\in (-\infty,0)$. Instead, the most reliable data in the range $c\in [-12.5, 0)$ is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for $c\in(-\infty,-12.5)$ display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.

Type

Publication

Classical and Quantum Gravity 36 (2019) 244001

Date

August, 2019

- E. Gwynne, “The dimension of the boundary of a Liouville quantum gravity metric ball”, arXiv preprint arXiv::1909.08588 (2019).
- Gwynne, Ewain. “Random surfaces and Liouville quantum gravity.” arXiv preprint arXiv:1908.05573 (2019).
- Gwynne E, Holden N, Pfeffer J, Remy G. Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach. Communications in Mathematical Physics.:1-53.
- Lehéricy, Thomas. “Cycles séparants, isopérimétrie et modifications de distances dans les grandes cartes planaires aléatoires.” Diss. Université Paris-Saclay, 2019.
- Kurov, Aleksandr, and Frank Saueressig. “On characterizing the Quantum Geometry underlying Asymptotic Safety.” arXiv preprint arXiv:2003.07454 (2020).
- Pfeffer, Joshua William. “Frontiers of Liouville quantum gravity.” PhD diss., Massachusetts Institute of Technology, 2020.
- Delporte, Nicolas. “Tensor Field Theories: Renormalization and Random Geometry.” arXiv preprint arXiv:2010.07819 (2020).
- Klitgaard, N. F. New curvatures for quantum gravity. Dissertation, Radboud University, Nijmegen, The Netherlands, 2022.
- Budd, Timothy, and Alicia Castro. “Scale-invariant random geometry from mating of trees: a numerical study.” arXiv preprint arXiv:2207.05355 (2022).
- Brunekreef, J. and Loll, R. “On the Nature of Spatial Universes in 3D Lorentzian Quantum Gravity”. arXiv preprint arXiv:2208.12718 (2022).