4.5 The phase structure

Already the first simulations by Agishtein and Migdal [2Jump To The Next Citation Point1Jump To The Next Citation Point] and Ambjørn and Jurkiewicz [11Jump To The Next Citation Point] of dynamically triangulated 4d gravity exhibited a clear two-phase structure. After tuning to the infinite-volume or critical line κc(κ ) 4 2, one identifies two regions, κ < κc 2 2 and κ > κc 2 2. The critical value c κ2 depends on the volume N4, and it was conjectured that it may move out to + ∞ as N4 → ∞ [75Jump To The Next Citation Point89Jump To The Next Citation Point], but it was later shown to converge to a finite value [14Jump To The Next Citation Point]. One can characterize the region with small κ2 < κc2 as the hot, crumpled, or condensed phase. It has small negative or positive curvature, large (possibly infinite) Hausdorff dimension dH and a high connectivity. By contrast, for κ > κc 2 2 one is in the cold, extended, elongated, or fluid phase. It has large positive curvature, with an effective tree-like branched-polymer geometry, and dH ≈ 2.

The location of the critical point on the infinite-volume line may be estimated from the peak in the curvature susceptibility χ(κ2,N4 ) = (⟨N 2⟩ − ⟨N2 ⟩2)∕N4 2 [2Jump To The Next Citation Point1Jump To The Next Citation Point11Jump To The Next Citation Point192Jump To The Next Citation Point129Jump To The Next Citation Point90Jump To The Next Citation Point], or the node susceptibility 2 2 χ(κ0,N4 ) = (⟨N0 ⟩ − ⟨N0 ⟩ )∕N4 [75Jump To The Next Citation Point76Jump To The Next Citation Point], as well as higher cumulants of N0 [48Jump To The Next Citation Point]. In [14Jump To The Next Citation Point], it was suggested that one may alternatively estimate c κ 2(N4 ) by looking at the behaviour of the entropy exponent γ, approaching κc2 from the elongated phase. More recently, Catterall et al. [77Jump To The Next Citation Point] have used the fluctuations χ0 = (⟨ω20⟩ − ⟨ω0 ⟩2)∕N4 of the local volume ω0 around singular vertices as an order parameter.


  Go to previous page Go up Go to next page