Connected Correlators in Quantum Gravity

We discuss the concept of connected, reparameterization invariant matter correlators in quantum gravity. We analyze the effect of discretization in two solvable cases : branched polymers and two-dimensional simplicial gravity. In both cases the naively defined connected correlators for a fixed volume display an anomalous behavior, which could be interpreted as a long-range order. We suggest that this is in fact only a highly non-trivial finite-size effect and propose an improved definition of the connected correlator, which reduces the effect. Using this definition we illustrate the appearance of a long-range spin order in the Ising model on a two-dimensional random lattice in an external magnetic field H , when H → 0 and β = βC . email [email protected] email [email protected] permanent address: Institute of Comp. Science, Jagellonian University, 30-072 Krakow, Poland email [email protected], permanent address: Institute of Physics, Jagellonian University, 30-059 Krakow, Poland