Curvature and scaling in 4D dynamical triangulation Original Research Article

We study the average number of simplices N′(r) at geodesic distance r in the dynamical triangulation model of euclidean quantum gravity in four dimensions. We use N′(r) to explore definitions of curvature and of effective global dimension. An effective curvature RV goes from negative values for low κ2 (the inverse bare Newton constant) to slightly positive values around the transition κc2. Far above the transition RV is hard to compute. This RV depends on the distance scale involved and we therefore investigate a similar explicitly r-dependent ‘running’ curvature Reff(r). This increases from values of order RV at intermediate distances to very high values at short distances. A global dimension d goes from high values in the region with low κ2 to d = 2 at high κ2. At the transition d is consistent with 4. We present evidence for scaling of N′(r) and introduce a scaling dimension ds which turns out to be approximately 4 in both weak and strong coupling regions. We discuss possible implications of the results, the emergence of classical euclidean spacetime and a possible “triviality” of the theory.