Catterall et al. [79, 80] observed that the pair of singular vertices form the end points of a singular link. They also offered a possible explanation for the formation of these singular structures: Simplices of sufficiently low dimension can maximize their local entropy by acquiring large local volumes (see also  for a mean-field argument). Catterall et al.  found two pseudo-critical points, and , associated with the creation of singular vertices and links, which seem to merge into a single critical point as . One concludes that the observed phase transition in the 4d dynamical triangulations model is driven by the appearance and disappearance of singular geometries.
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