Multigraph models for causal quantum gravity and scale dependent spectral dimension

Abstract

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 ρ, 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.

Standard Model in multiscale theories and observational constraints

“…The transition between the two regimes varies depending on the model but it is continuous in general. Examples include causal dynamical triangulations [5][6][7], asymptotically safe quantum gravity [8,9], loop quantum gravity and spin foams [10][11][12], Hořava-Lifshitz gravity [7,9,13], noncommutative geometry [14][15][16] and κ-Minkowski spacetime [17,18], nonlocal quantum gravity [19], Stelle's gravity [20], spacetimes with black holes [21][22][23], fuzzy spacetimes [24], random combs [25,26], random multigraphs [27,28], and causal sets [29].…”

“…Similarly, in the integer picture the multiscale theory with weighted derivatives is not general relativity with minimally coupled matter, and one can never trivialize the theory to the ordinary one as in the flat case (28). The gravitational dynamics of the theory with weighted derivatives was studied in [43].…”

Diffusion in multiscale spacetimes

“…(d) Random multigraphs, whose geometry was shown to be closely akin to (certain approximations of) causal dynamical triangulations [8,9]. The analytic form of the return probability is, in fact,…”

“…The normalized analytic solution can be written in different ways [68]. For us, the following is 9 A local extremum, for instance a minimum value d S,min at some point 0 < min < +∞, would signal the presence of another scale because this feature could not be removed by a finite conformal rescaling of diffusion time. 10 Some remarks on the case with a finite but arbitrary number N of diffusion operators are given in Ref.…”

“…In general, one of the indicators characterizing quantum geometry, the spectral dimension d S of spacetime, changes with the scale, running from d S 2 (or exactly d S = 2) in the ultraviolet (UV) to the usual, classical value d S ∼ 4 in the infrared (IR). Numerical and analytic examples can be found in causal dynamical triangulations (CDT) [4,5], random combs [6,7] and random multigraphs [8,9] (both sharing some properties with CDT), quantum Einstein gravity (QEG, also called asymptotic safety) [10,11], spin foams [12][13][14][15], Hořava-Lifshitz gravity [16,17], noncommutative geometry at the fundamental [18,19] and effective [20][21][22] levels, field theory on multifractal spacetimes [23][24][25] (in particular, in the realization within multifractional geometry [26][27][28][29][30][31]), and nonlocal super-renormalizable quantum gravity [32][33][34].…”