Approaches to Quantum Gravity

350 J. Ambjørn, J. Jurkiewicz and R. Loll0.80.6(V 3 (0)V 3 (δ))0.40.20–4 –2 0 2 4xFig. 18.3. The scaling of the volume–volume correlator, as function of therescaled time variable x = δ/(N 4 ) 1/4 . Data points come from system sizes N 4 =22 500, 45 000, 91 000, 181 000 and 362 000 at κ 0 =2.2, =0.6 and T =80.on the underlying geometric ensemble. On a d-dimensional manifold with a fixed,smooth Riemannian metric g ab (ξ), the diffusion equation has the form∂∂σ K g(ξ, ξ 0 ; σ) = g K g (ξ, ξ 0 ; σ), (18.11)where σ is a fictitious diffusion time, g the Laplace operator of the metric g ab (ξ)and K g (ξ, ξ 0 ; σ) the probability density of diffusion from point ξ 0 to point ξ indiffusion time σ . We will consider diffusion processes which initially are peakedat some point ξ 0 ,sothatK g (ξ, ξ 0 ; σ =0) =For the special case of a flat Euclidean metric, we have1√ detg(ξ)δ d (ξ − ξ 0 ). (18.12)K g (ξ, ξ 0 ; σ) = e−d2 g (ξ,ξ 0)/4σ(4πσ) d/2 , g ab(ξ)=δ ab , (18.13)where d g denotes the distance function associated with the metric g.A quantity which is easier to measure in numerical simulations is the averagereturn probability P g (σ ), definedbyP g (σ ) := 1 ∫d d ξ √ detg(ξ) K g (ξ, ξ; σ), (18.14)Vwhere V is the spacetime volume V = ∫ d d ξ √ detg(ξ). For an infinite flat space,we have P g (σ ) = 1/(4πσ) d/2 and thus can extract the dimension d by taking thelogarithmic derivative− 2 d log P g(σ )d log σ= d, (18.15)