4.6 Evidence for a second-order 4 Dynamical triangulations4.4 Performing the state sum

4.5 The phase structure

Already the first simulations by Agishtein and Migdal [2Jump To The Next Citation Point In The Article, 1Jump To The Next Citation Point In The Article] and Ambjørn and Jurkiewicz [11Jump To The Next Citation Point In The Article] of dynamically triangulated 4d gravity exhibited a clear two-phase structure. After tuning to the infinite-volume or critical line tex2html_wrap_inline2967, one identifies two regions, tex2html_wrap_inline2969 and tex2html_wrap_inline2971 . The critical value tex2html_wrap_inline2973 depends on the volume tex2html_wrap_inline2863, and it was conjectured that it may move out to tex2html_wrap_inline2977 as tex2html_wrap_inline2907 [75Jump To The Next Citation Point In The Article, 89Jump To The Next Citation Point In The Article], but it was later shown to converge to a finite value [14Jump To The Next Citation Point In The Article]. One can characterize the region with small tex2html_wrap_inline2969 as the hot, crumpled, or condensed phase. It has small negative or positive curvature, large (possibly infinite) Hausdorff dimension tex2html_wrap_inline2983 and a high connectivity. By contrast, for tex2html_wrap_inline2971 one is in the cold, extended, elongated, or fluid phase. It has large positive curvature, with an effective tree-like branched-polymer geometry, and tex2html_wrap_inline2987 .

The location of the critical point on the infinite-volume line may be estimated from the peak in the curvature susceptibility tex2html_wrap_inline2989 [2Jump To The Next Citation Point In The Article, 1Jump To The Next Citation Point In The Article, 11Jump To The Next Citation Point In The Article, 192Jump To The Next Citation Point In The Article, 129Jump To The Next Citation Point In The Article, 90Jump To The Next Citation Point In The Article], or the node susceptibility tex2html_wrap_inline2991 [75Jump To The Next Citation Point In The Article, 76Jump To The Next Citation Point In The Article], as well as higher cumulants of tex2html_wrap_inline2891 [48Jump To The Next Citation Point In The Article]. In [14Jump To The Next Citation Point In The Article], it was suggested that one may alternatively estimate tex2html_wrap_inline2995 by looking at the behaviour of the entropy exponent tex2html_wrap_inline2959, approaching tex2html_wrap_inline2973 from the elongated phase. More recently, Catterall et al [77Jump To The Next Citation Point In The Article] have used the fluctuations tex2html_wrap_inline3001 of the local volume tex2html_wrap_inline3003 around singular vertices as an order parameter.



4.6 Evidence for a second-order 4 Dynamical triangulations4.4 Performing the state sum

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