4.11 Singular configurations

The singular nature of the geometry in the phase below the critical value c κ2 can be quantified by the distribution ρ (n ) of the vertex order o(v),
⟨∑ ⟩ ρ(n ) = 1-- δo(v),n , (24 ) N0 v
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 ∑ ∼ v(o(v) − 5 )2 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. [7980] 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, (1) κ0 and (2) κ 0, associated with the creation of singular vertices and links, which seem to merge into a single critical point κc0 as V → ∞. 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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