4.14 Two-point functions

It is possible to define two-point correlation functions on random geometries [1614], which are to be thought of as the discrete analogues of formal continuum correlators
∫ ∫ ∫ G (r) = Dg e−S d4x d4x′∘det--g(x)∘det-g-(x′)𝒪 (x)𝒪 (x′)δ(d (x, x′) − r), (25 ) 𝒪 RiemS4 μν g DiffS4
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 (25View Equation), as was pointed out by de Bakker and Smit [9192]. 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.
  Go to previous page Go up Go to next page