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 R V goes from negative values for low κ 2 (the inverse bare Newton constant) to slightly positive values around the transition κ c 2 . Far above the transition R V is hard to compute. This R V depends on the distance scale involved and we therefore investigate a similar explicitly r dependent 'running' curvature R eff (r). This increases from values of order R V at intermediate distances to very high values at short distances. A global dimension d goes from high values in the region with low κ 2 to d = 2 at high κ 2 . At the transition d is consistent with 4. We present evidence for scaling of N ′ (r) and introduce a scaling dimension d s 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. *

We confirm recent claims that, contrary to what was generally believed, the phase transition of the dynamical triangulation model of four-dimensional quantum gravity is of first order. We have looked at this at a volume of 64, 000 four-simplices, where the evidence in the form of a double peak histogram of the action is quite clear.

In the dynamical triangulation model of four dimensional euclidean quantum gravity we investigate gravitational binding. Two scalar test particles (quenched approximation) have a positive binding energy, thereby showing that the model can represent gravitational attraction.Comment: 19 pages, LaTeX2e, version as accepted by Nucl Phys

In the dynamical triangulation model of 4D euclidean quantum gravity we measure two-point functions of the scalar curvature as a function of the geodesic distance. To get the correlations it turns out that we need to subtract a squared one-point function which, although this seems paradoxical, depends on the distance. At the transition and in the elongated phase we observe a power law behaviour, while in the crumpled phase we cannot find a simple function to describe it.Comment: 16 pages, LaTeX2e source with postscript result, version to be published in Nucl. Phys.