Singular configurations

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 $o(v)$ ,

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. [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, $(1) κ0$ 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.