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 $-\infty$. 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.