From Trees to Gravity

TL;DR

This paper investigates quantum geometric models, specifically random trees and causal triangulations, calculating their dimensions and exploring how matter interactions influence their structure using probabilistic and graph theory methods.

Contribution

It provides new calculations of Hausdorff and spectral dimensions for these models and explores their relationship with the underlying geometry, including effects of matter fields.

Findings

Hausdorff and spectral dimensions are computed for the models

The relationship between geometry and dimensions is analyzed

Interactions with matter fields are briefly discussed

Abstract

In this article we study two related models of quantum geometry: generic random trees and two-dimensional causal triangulations. The Hausdorff and spectral dimensions that arise in these models are calculated and their relationship with the structure of the underlying random geometry is explored. Modifications due to interactions with matter fields are also briefly discussed. The approach to the subject is that of classical statistical mechanics and most of the tools come from probability and graph theory.