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 cϵ(-∞, 0). Instead, the most reliable data in the range c∞[-12.5) is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for cϵ(-∞-12.5) display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.
| Original language | English |
|---|---|
| Article number | 244001 |
| Journal | Classical and Quantum Gravity |
| Volume | 36 |
| Issue number | 24 |
| Number of pages | 25 |
| ISSN | 0264-9381 |
| DOIs | |
| Publication status | Published - 14 Nov 2019 |
Keywords
- 2D quantum gravity
- Liouville quantum gravity
- random planar maps
- fractal dimensions
- Monte Carlo simulation