The phase structure

4.5 The phase structure

Already the first simulations by Agishtein and Migdal [2

, 1

] and Ambjørn and Jurkiewicz [11

] 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 $N4 → ∞$ [75

, 89

], but it was later shown to converge to a finite value [14

]. 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 $χ(κ2,N4 ) = (⟨N 2⟩ − ⟨N2 ⟩2)∕N4 2$ [2 , 1 , 11 , 192 , 129 , 90 ], or the node susceptibility $2 2 χ(κ0,N4 ) = (⟨N0 ⟩ − ⟨N0 ⟩ )∕N4$ [75 , 76 ], as well as higher cumulants of $N0$ [48 ]. In [14 ], 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. [77 ] have used the fluctuations $χ0 = (⟨ω20⟩ − ⟨ω0 ⟩2)∕N4$ of the local volume $ω0$ around singular vertices as an order parameter.