The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

TL;DR

This paper investigates the geometric properties of 2D Euclidean quantum gravity on a torus, focusing on shortest non-contractible loops, and finds that their length distribution aligns with Watabiki's Hausdorff dimension formula.

Contribution

It provides numerical evidence that the Hausdorff dimension of 2D Euclidean quantum gravity on a torus matches Watabiki's theoretical prediction.

Findings

Distribution of geodesic lengths scales according to Watabiki's Hausdorff dimension

Numerical results support the theoretical formula for Hausdorff dimension

Study focuses on quantum gravity coupled to conformal field theories with c<1

Abstract

The lengths of shortest non-contractible loops are studied numerically in 2d Euclidean quantum gravity on a torus coupled to conformal field theories with central charge less than one. We find that the distribution of these geodesic lengths displays a scaling in agreement with a Hausdorff dimension given by the formula of Y. Watabiki.