SCALING IN QUANTUM-GRAVITY

The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential falloff of the 2-point function with geodesic distance determines the fractal dimension d(H) of space-time. The integral of the 2-point function determines the entropy exponent gamma, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects gamma and d(H) by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation.