### 4.14 Two-point functions

It is possible to define two-point correlation functions on random geometries [16, 14], which are to be
thought of as the discrete analogues of formal continuum correlators
for local observables , with denoting the geodesic distance with respect to the metric .
There is an ambiguity in defining the connected part of the correlator (25), as was pointed out by
de Bakker and Smit [91, 92]. Contrary to expectations, after subtraction of the square of the curvature
expectation value, the resulting quantity does not scale to zero with large distances. They
therefore proposed an alternative definition of the connected two-point function, by subtracting the
square of a “curvature-to-nothing” correlator . This definition was compared in more
detail by Bialas [45] with a more conventional notion, as, for example, the one used in [48].
For the case of curvature correlators, their behaviour differs significantly, especially at short
distances.