### 4.11 Singular configurations

The singular nature of the geometry in the phase below the critical value can be quantified by the
distribution of the vertex order ,
first considered by Hotta et al. [129]. It has a continuum part and a separate peak at high vertex order. One
can suppress this effect by adding a term to the action (see also [19] for a discussion of
terms of a similar nature), but this leads to a simultaneous disappearance of the phase transition. Hotta et
al. [130] have checked that for a variety of initial configurations the singular structure is a generic feature of
the model.
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 [49]
for a mean-field argument). Catterall et al. [77] 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.