Multigraph models for causal quantum gravity and scale dependent spectral dimension

TL;DR

This paper investigates the spectral dimension of multigraph models related to quantum gravity, revealing scale-dependent behaviors and conditions for equality with Hausdorff dimension, with implications for models of causal dynamical triangulation.

Contribution

It introduces the concept of scale dependent spectral dimension in multigraph models and applies it to quantum gravity, showing how spectral dimension varies with scale in these models.

Findings

Spectral dimension equals Hausdorff dimension when resistance exponent is zero.

Scale dependent spectral dimension is demonstrated with values two at large scales and one at small scales.

A specific model related to four-dimensional CDT exhibits spectral dimension four at large scales and two at small scales.

Abstract

We study random walks on ensembles of a specific class of random multigraphs which provide an "effective graph ensemble" for the causal dynamical triangulation (CDT) model of quantum gravity. In particular, we investigate the spectral dimension of the multigraph ensemble for recurrent as well as transient walks. We investigate the circumstances in which the spectral dimension and Hausdorff dimension are equal and show that this occurs when rho, the exponent for anomalous behaviour of the resistance to infinity, is zero. The concept of scale dependent spectral dimension in these models is introduced. We apply this notion to a multigraph ensemble with a measure induced by a size biased critical Galton-Watson process which has a scale dependent spectral dimension of two at large scales and one at small scales. We conclude by discussing a specific model related to four dimensional CDT which has a spectral dimension of four at large scales and two at small scales.