### 4.6 Evidence for a second-order transition?

It sometimes seems to be assumed that if one were to find a continuum theory at a second-order phase
transition, it would have flat Minkowski space as its ground state (in spite of the -topology), and
gravitonic spin-2 excitations. An alternative scenario with a constant-curvature sphere-metric has been put
forward by de Bakker and Smit [90, 92].
Agishtein and Migdal [2] initially reported a hysteresis in the average curvature , indicating a
first-order transition. However, subsequent authors [11, 191, 192, 75, 76] found numerical
data not incompatible with the existence of a second-order transition, and also Agishtein and
Migdal [1] retracted their original claim as a result of a closer examination of the fixed point
region.

One tries to discriminate between a first- and second- (or higher-)order transition by looking at the
Binder parameter [191, 192, 7], or scaling exponents governing the scaling behaviour of
suitable observables [1, 192], or the peak height of susceptibilities as a function of the volume
[16, 75, 76, 14, 48, 87]. Other scaling relations are discussed in [90, 91, 96, 95]. However, since
critical parameters are hard to measure, and it is difficult to estimate finite-size effects, none of the data can
claim to be conclusive.

Some doubts were cast on the conjectured continuous nature of the phase transition by
Bialas et al. [48], who found an unexpected two-peak structure in the distribution of nodes
near the fixed point. This was strengthened further by data taken at 64k by de Bakker [87],
with an even more pronounced double peak (see also [51]). Most likely previous simulations
were simply too small to detect the true nature of the phase transition. Both Bialas et al. and
de Bakker observed that the finite size scaling exponents extracted from the node susceptibility
grow with volume and may well reach the value 1 expected for a first-order transition as
.

The origin of this behaviour was further elucidated by Bialas et al. [49, 47] and Bialas and Burda [46],
who found a simple mean-field model that reproduces qualitatively the phase structure of 4d dynamically
triangulated quantum gravity. With an appropriate choice of local weights, this model has a condensed and
a fluid phase, with a first-order transition in between. A similar behaviour was found by Catterall et
al. [77], who made a related mean-field ansatz, with the local weights depending on the local entropies
around the vertices.