Scaling and the Fractal Geometry of Two-Dimensional Quantum Gravity

TL;DR

This paper investigates the scaling behavior of geodesic correlations in 2D quantum gravity, revealing a non-perturbative length scale and estimating the Hausdorff dimension, supporting theoretical predictions.

Contribution

It provides numerical evidence for a non-perturbative length scale in 2D quantum gravity and estimates the Hausdorff dimension, aligning with recent theoretical results.

Findings

Hausdorff dimension approximately 3.8

Supports d_H = 4 theoretical prediction

Identifies a non-perturbative length scale in quantum geometry

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.