### 3.3 First simulations

The first numerical studies of the Regge action were undertaken by Berg [38, 39] and Hamber and
Williams [118, 119]. Berg performed a Monte Carlo simulation of the pure-curvature action for hypercubic
2^{4}- and 3^{4}-lattices with simplicial subdivision (see [40] for a description of the method). He used the
scale-invariant measure . To avoid the divergence that results from rescaling the link lengths,
he kept the total volume constant by performing an overall length rescaling of all links after each move.
This amounts to fixing a typical length scale , where is the expectation value of the
4-simplex volume.
For , he found a negative average curvature , and some evidence for a canonical scaling
behaviour of lengths, areas and volumes. For , he obtained a negative (positive) average deficit
angle and a positive (negative) . A more detailed analysis for led Berg [39] to
conclude that there exists a critical value (presumably a first-order transition [40]), below which
is convergent, whereas above it diverges. (Myers [169] has conjectured that it may be possible to perform a
similar analysis for Monte Carlo data for the Lorentzian action.)

By contrast, Hamber and Williams [118] simulated the higher-derivative action

on 2^{4}- and 4^{4}-lattices, using a time-discretized form of the Langevin evolution equation (see also [106]). For
technical reasons, one uses barycentric instead of Voronoi volumes. They employed the scale-invariant
measure , where enforces an ultra-violet cutoff . They investigated the average
curvature and squared curvature (scaled by powers of to make them dimensionless),
as well as and . For , one finds a negative and a large ,
indicating a rough geometry. For small , one observes a sudden decrease in , as well as a jump
from large positive to small negative values of as is increased. For large ,
is small and negative, and the geometry appears to be smooth. Like Berg, they advocated a
fundamental-length scenario, where the dynamically determined average link length provides an effective
UV-cutoff.