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.
![]() |
Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |