4.12 Renormalization group4 Dynamical triangulations4.10 Non-spherical lattices

4.11 Singular configurations

The singular nature of the geometry in the phase below the critical value tex2html_wrap_inline2973 can be quantified by the distribution tex2html_wrap_inline3079 of the vertex order o (v),

  equation836

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 tex2html_wrap_inline3083 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, tex2html_wrap_inline3085 and tex2html_wrap_inline3087, associated with the creation of singular vertices and links, which seem to merge into a single critical point tex2html_wrap_inline3089 as tex2html_wrap_inline3017 . One concludes that the observed phase transition in the 4d dynamical triangulations model is driven by the appearance and disappearance of singular geometries.



4.12 Renormalization group4 Dynamical triangulations4.10 Non-spherical lattices

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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