### 3.8 Two-point functions

To understand the nature of the possible excitations at the phase transition, one needs to study
correlation functions in the vicinity of , which is difficult numerically. Some data are available on the
connected correlation functions of the curvatures and the volumes at fixed geodesic distance ,
and , for lattice sizes 16^{4} [115], using a scalar field propagator to determine . Both
correlators were measured at various -values, leading to similar results for both and
. The data, taken for ( lattice spacings), can be fitted to decaying
exponentials.
Some further data (for ) were reported by the Vienna group [37, 34]. These authors simply used
the lattice distance instead of the true geodesic distance . In [37], the measure was taken
to be of the form . They looked at on - and -lattices, for
and , and found a fast decay for all investigated values of , and
.