3.4 The phase structure

Further evidence for a transition between a region of rough and smooth geometry comes from Monte Carlo simulations by Hamber [108111Jump To The Next Citation Point] on 44- and 84-lattices, this time with the lattice ∏ 𝒟l = ilidli (see also [114] for a summary of results, and [109] for more details on the method). There is a value kc at which the average curvature vanishes. For k > k c, the curvature becomes large and the simplices degenerate into configurations with very small volumes. He performed a simultaneous fit for kc, A ℛ and δ in the scaling relation
k→kc δ ℛ (k) ∼ A ℛ(kc − k) . (17 )
This leads to a scaling exponent δ ≈ 0.60, with only a weak dependence on a. There are even points with a = 0 that lie in the well-defined, smooth phase. Hamber also investigated the curvature and volume susceptibilities χ ℛ and χV. At a continuous phase transition, χℛ should diverge, reflecting long-range correlations of a massless graviton excitation. The data obtained are not incompatible with such a scenario, but the extrapolation to the transition point kc seems somewhat ambiguous. On the other hand, one does not expect χ V to diverge at k c, which is corroborated by the simulations.
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