### 3.9 Non-hypercubic lattices

Simulations on lattices with irregular link geometry (still with -topology) have been performed by
Beirl et al. [31, 36]. They were obtained by adding a few vertices of low coordination number to otherwise
regular lattices. One finds that the average curvature increases from negative to positive values, even
for , as a result of the formation of spikes. In [36], the averages were monitored separately at
the regular and the inserted vertices.
The dependence of on is rather interesting: One observes two “critical” points, a smaller
one , where the extra vertices develop spikes, and a second one where the remaining
vertices follow. undergoes a small jump at , and a larger one at . There is also a
transition point to a phase with collapsed simplices at large negative , with a jump to large
negative . Additional transition points at negative were also found in simulations of the
“compactified” Regge action (this action was discussed in [104];
see also [71, 140]) and a -version of Regge gravity [33]. Correlation functions at those
points were computed in [34] for short distances, but no evidence for long-range correlations was
found.

The same authors studied the inclusion of the higher-derivative term (15) in [36]. On the regular lattice,
their findings for confirmed those by Hamber, apart from the fact that they found stable
expectation values even for positive . Inserting irregular vertices pushes to larger values and leads
again to the appearance of an additional transition point.