### 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 4^{4}- and 8^{4}-lattices, this time with the lattice
(see also [114] for a summary of results, and [109] for more details on the method). There is a value at
which the average curvature vanishes. For , the curvature becomes large and the simplices
degenerate into configurations with very small volumes. He performed a simultaneous fit for , and
in the scaling relation
This leads to a scaling exponent , with only a weak dependence on . There are even points
with that lie in the well-defined, smooth phase. Hamber also investigated the curvature and volume
susceptibilities and . At a continuous phase transition, 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 seems somewhat ambiguous. On
the other hand, one does not expect to diverge at , which is corroborated by the
simulations.