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.
| Originalsprog | Engelsk |
|---|---|
| Artikelnummer | 244001 |
| Tidsskrift | Classical and Quantum Gravity |
| Vol/bind | 36 |
| Udgave nummer | 24 |
| Antal sider | 25 |
| ISSN | 0264-9381 |
| DOI | |
| Status | Udgivet - 14 nov. 2019 |