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 ; see also [71, 140]) and a -version of Regge gravity . Correlation functions at those points were computed in  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 . 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.
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