Multigraph models for causal quantum gravity and scale dependent spectral dimension. (English) Zbl 1251.83021
Summary: 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.
MSC:
83C45 | Quantization of the gravitational field |
83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
62P35 | Applications of statistics to physics |
54F65 | Topological characterizations of particular spaces |
81T20 | Quantum field theory on curved space or space-time backgrounds |