Two-point functions

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 $kc$ , which is difficult numerically. Some data are available on the connected correlation functions of the curvatures and the volumes at fixed geodesic distance $d$ , $G (d) R$ and $GV (d)$ , for lattice sizes $≤$ 16⁴ [115 ], using a scalar field propagator to determine $d$ . Both correlators were measured at various $k$ -values, leading to similar results for both $a = 0$ and $a = 0.005$ . The data, taken for $d ≤ 16$ ( $≃ 7$ lattice spacings), can be fitted to decaying exponentials.

Some further data (for $a = 0$ ) were reported by the Vienna group [37 , 34 ]. These authors simply used the lattice distance $n$ instead of the true geodesic distance $d$ . In [37 ], the measure was taken to be of the form $l2σ− 1dl$ . They looked at $GV (n)$ on $33 × 8$ - and $43 × 16$ -lattices, for $λ = σ = 1$ and $λ = σ = 0.1$ , and found a fast decay for all investigated values of $k$ , and $n ≤ 8$ .