Scaling and the Fractal Geometry of Two-Dimensional Quantum Gravity

TL;DR

This work investigates the fractal geometry of two-dimensional quantum gravity by examining geodesic correlation functions on dynamical triangulations. Using finite-size scaling, the authors identify a non-perturbative length scale $l_G$ and extract the Hausdorff dimension $d_H$, finding $d_H \approx 3.83$ for pure gravity, in line with the theoretical value $d_H=4$, and observe broadly similar $d_H$ when simple spins are coupled to gravity, with limited evidence of back-reaction within current finite-size limits. Analytic discussions of $d_H$ from continuum Liouville theory and matrix models reveal multiple proposed formulas that sometimes disagree for certain minimal models, underscoring ambiguities in defining fractal dimensions on fluctuating geometries. The results support a scaling picture with a dynamically generated length scale, but also motivate a two-scale hypothesis and further high-resolution simulations to resolve potential back-reaction effects and model-dependent discrepancies.

Abstract

We examine the scaling of geodesic correlation functions in two-dimensional gravity and in spin systems coupled to gravity. The numerical data support the scaling hypothesis and indicate that the quantum geometry develops a non-perturbative length scale. The existence of this length scale allows us to extract a Hausdorff dimension. In the case of pure gravity we find d_H approx. 3.8, in support of recent theoretical calculations that d_H = 4. We also discuss the back-reaction of matter on the geometry.

Figures (4)

• Figure 1: Volume scaling of (${\it a}$) the location of the peak $r_0$ in the distributions $n(r,N)$ and ( b) their maximal value $n(r_0)$ in the case of pure gravity. Data is shown both for the direct and dual lattices and the extracted values of $d_H$ are included.
• Figure 2: Scaling plots for the point-point distributions $n(r,N)$ in the case of pure gravity; (a) the direct and (b) dual lattice. Shown are the curves fitted to distributions after rescaling. The value of $d_H$ is chosen so as it minimized the total chi-square of the fits.
• Figure 3: Volume scaling for $r_0$ and $n(r_0)$ for the distributions we measured for Ising model coupled to gravity. The same scaling behavior is used to extract $d_H$ from the slope as in the case of pure gravity, except for $g_{un}(r_0)$. There we used $n(r_0)\sim \gamma/\nu d_H - 1/d_H$, substituting the exact values for $\gamma/\nu d_H$.
• Figure 4: Collapsing the data for $n(r,N)$ and $g_{un}(r,N)$ on a single curve using one scaling parameter in the case of an Ising model coupled to gravity.