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 [37Jump To The Next Citation Point34Jump To The Next Citation Point]. 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.

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