Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
Abstract
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 . 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 . Instead, the most reliable data in the range is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.
- Publication:
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Classical and Quantum Gravity
- Pub Date:
- December 2019
- DOI:
- 10.1088/1361-6382/ab4f21
- arXiv:
- arXiv:1908.09469
- Bibcode:
- 2019CQGra..36x4001B
- Keywords:
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- General Relativity and Quantum Cosmology;
- Mathematical Physics
- E-Print:
- 26 pages, 15 figures