### 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 , one identifies two regions, and . The
critical value depends on the volume , and it was conjectured that it may move out to
as [75, 89], but it was later shown to converge to a finite value [14]. One can
characterize the region with small as the hot, crumpled, or condensed phase. It has
small negative or positive curvature, large (possibly infinite) Hausdorff dimension and a
high connectivity. By contrast, for one is in the cold, extended, elongated, or fluid
phase. It has large positive curvature, with an effective tree-like branched-polymer geometry, and
.
The location of the critical point on the infinite-volume line may be estimated from the peak in the
curvature susceptibility [2, 1, 11, 192, 129, 90], or the node
susceptibility [75, 76], as well as higher cumulants of [48]. In [14],
it was suggested that one may alternatively estimate by looking at the behaviour of the entropy
exponent , approaching from the elongated phase. More recently, Catterall et al. [77] have used the
fluctuations of the local volume around singular vertices as an order
parameter.