Abstract
In the weak field expansion of euclidean quantum gravity, an analysis of the Wilson loops in terms of the gauge group, SO(4), shows that the correspondent statistical system does not develop any configuration with localized curvature at low temperature. Such a ''non-local'' behavior contrasts strongly with that of usual gauge fields. We point out a possible relation between this result and those of the numerical simulations of quantum Regge Calculus. We also give a general quantum formula for the static potential energy of the gravitational interaction of two masses, which is compatible with the mentioned vacuum structure.