Two-point functions

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 $𝒪 (x )$ , with $dg$ denoting the geodesic distance with respect to the metric $gμν$ . 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 $2 ⟨RR ⟩(r) − ⟨R ⟩$ 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 $⟨R ⟩(r)$ . 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.