3.9 Non-hypercubic lattices

Simulations on lattices with irregular link geometry (still with T 4-topology) have been performed by Beirl et al. [3136Jump To The Next Citation Point]. They were obtained by adding a few vertices of low coordination number to otherwise regular lattices. One finds that the average curvature ⟨R⟩ increases from negative to positive values, even for k < kc, as a result of the formation of spikes. In [36Jump To The Next Citation Point], the averages 2 ⟨l⟩ were monitored separately at the regular and the inserted vertices.

The dependence of ℛ on k is rather interesting: One observes two “critical” points, a smaller one k1, where the extra vertices develop spikes, and a second one k2 where the remaining vertices follow. ℛ undergoes a small jump at k 1, and a larger one at k 2. There is also a transition point to a phase with collapsed simplices at large negative k, with a jump to large negative ℛ. Additional transition points at negative k were also found in simulations of the “compactified” Regge action ∑ S = b(− k Ab sinδb + λVb) (this action was discussed in [104Jump To The Next Citation Point]; see also [71Jump To The Next Citation Point140Jump To The Next Citation Point]) and a ℤ2-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 (15View Equation) in [36]. On the regular lattice, their findings for ℛ (k,a ) confirmed those by Hamber, apart from the fact that they found stable expectation values even for positive ℛ. Inserting irregular vertices pushes ℛ to larger values and leads again to the appearance of an additional transition point.


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