1202.6322 (Georgios Giasemidis et al.)

Multigraph models for causal quantum gravity and scale dependent
spectral dimension    [PDF]

Georgios Giasemidis, John F Wheater, Stefan Zohren

We study random walks on ensembles of a specific class of random multigraph graphs associated with theories of causal quantum gravity. In particular, we investigate the spectral dimension of the graph 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 quantum gravity which has a spectral dimension of four at large scales and two at small scales.

View original: http://arxiv.org/abs/1202.6322