Approaches to Quantum Gravity

352 J. Ambjørn, J. Jurkiewicz and R. Loll3.83.63.4D S3.232.80 100 200 300 400σFig. 18.4. The spectral dimension D S of the universe as a function of the diffusiontime σ , measured for κ 0 = 2.2, = 0.6 and t = 80, and a spacetime volumeN 4 =181k. The averaged measurements lie along the central curve, together witha superimposed best fit D S (σ ) = 4.02−119/(54+σ)(thin black curve). The twoouter curves represent error bars.as long as the diffusion time is not much larger than N 2/D S4. The outcome of themeasurements is presented in Fig. 18.4, with error bars included. (The two outercurves represent the envelopes to the tops and bottoms of the error bars.) The errorgrows linearly with σ , due to the presence of the log σ in (18.21).The remarkable feature of the curve D S (σ ) is its slow approach to the asymptoticvalue of D S (σ ) for large σ . The new phenomenon we observe here is a scaledependence of the spectral dimension, which has emerged dynamically [11; 10].As explained by [11], the best three-parameter fit which asymptoticallyapproaches a constant is of the formD S (σ ) = a −b119= 4.02 −σ + c 54 + σ . (18.22)The constants a, b and c have been determined by using the data range σ ∈[40, 400] and the curve shape agrees well with the measurements, as can be seenfrom Fig. 18.4.Integrating(18.22) we obtain1P(σ ) ∼, (18.23)σ a/2 (1 + c/σ )b/2cfrom which we deduce the limiting cases⎧⎨ σ −a/2 for large σ ,P(σ ) ∼(18.24)⎩σ −(a−b/c)/2 for small σ.