# Phases and fractal structures of three-dimensional simplicial gravity

## TL;DR

This paper investigates the different phases and fractal geometries of three-dimensional simplicial quantum gravity using Monte Carlo simulations, revealing distinct structural properties and Hausdorff dimensions across phases.

## Contribution

It classifies the phases of 3D simplicial quantum gravity and characterizes their fractal structures and Hausdorff dimensions through surface area distribution analysis.

## Key findings

- Crumpled phase with Hausdorff dimension ~5 and complex topology.
- Critical point phase exhibits fractal-like manifold with dimension ~4.
- Cold phase resembles branched-polymer with dimension ~2 and spherical cross-sections.

## Abstract

We study phases and fractal structures of three-dimensional simplicial quantum gravity by the Monte-Carlo method. After measuring the surface area distribution (SAD) which is the three-dimensional analog of the loop length distribution (LLD) in two-dimensional quantum gravity, we classify the fractal structures into three types: (i) in the hot (strong coupling) phase, strong gravity makes the space-time one crumpled mother universe with small fluctuating branches around it. This is a crumpled phase with a large Hausdorff dimension $d_{\mbox{\tiny H}} \simeq 5$. The topologies of cross-sections are extremely complicated. (ii) at the critical point, we observe that the space-time is a fractal-like manifold which has one mother universe with small and middle size branches around it. The Hausdorff dimension is $d_{\mbox{\tiny H}} \simeq 4$. We observe some scaling behaviors for the cross-sections of the manifold. This manifold resembles the fractal surface observed in two-dimensional quantum gravity. (iii) in the cold (weak coupling) phase, the mother universe disappears completely and the space-time seems to be the branched-polymer with a small Hausdorff dimension $d_{\mbox{\tiny H}} \simeq 2$. Almost all of the cross-sections have the spherical topology $S^2$ in the cold phase.