Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity


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.

Classical and Quantum Gravity 36 (2019) 244001


  1. E. Gwynne, “The dimension of the boundary of a Liouville quantum gravity metric ball”, arXiv preprint arXiv::1909.08588 (2019).
  2. Gwynne, Ewain. “Random surfaces and Liouville quantum gravity.” arXiv preprint arXiv:1908.05573 (2019).
  3. 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.
  4. Lehéricy, Thomas. “Cycles séparants, isopérimétrie et modifications de distances dans les grandes cartes planaires aléatoires.” Diss. Université Paris-Saclay, 2019.
  5. Kurov, Aleksandr, and Frank Saueressig. “On characterizing the Quantum Geometry underlying Asymptotic Safety.” arXiv preprint arXiv:2003.07454 (2020).
  6. Pfeffer, Joshua William. “Frontiers of Liouville quantum gravity.” PhD diss., Massachusetts Institute of Technology, 2020.