The phase structure

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 [108 , 111

] 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

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.