4.7 Influence of the measure4 Dynamical triangulations4.5 The phase structure

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 tex2html_wrap_inline2837 -topology), and gravitonic spin-2 excitations. An alternative scenario with a constant-curvature sphere-metric has been put forward by de Bakker and Smit [90Jump To The Next Citation Point In The Article, 92Jump To The Next Citation Point In The Article].

Agishtein and Migdal [2] initially reported a hysteresis in the average curvature tex2html_wrap_inline3007, indicating a first-order transition. However, subsequent authors [11Jump To The Next Citation Point In The Article, 191Jump To The Next Citation Point In The Article, 192Jump To The Next Citation Point In The Article, 75Jump To The Next Citation Point In The Article, 76Jump To The Next Citation Point In The Article] found numerical data not incompatible with the existence of a second-order transition, and also Agishtein and Migdal [1Jump To The Next Citation Point In The Article] 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, 192Jump To The Next Citation Point In The Article, 7Jump To The Next Citation Point In The Article], or scaling exponents tex2html_wrap_inline3009 governing the scaling behaviour tex2html_wrap_inline3011 of suitable observables [1, 192], or the peak height of susceptibilities as a function of the volume tex2html_wrap_inline2863 [16Jump To The Next Citation Point In The Article, 75Jump To The Next Citation Point In The Article, 76Jump To The Next Citation Point In The Article, 14Jump To The Next Citation Point In The Article, 48Jump To The Next Citation Point In The Article, 87Jump To The Next Citation Point In The Article]. Other scaling relations are discussed in [90Jump To The Next Citation Point In The Article, 91Jump To The Next Citation Point In The Article, 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 [48Jump To The Next Citation Point In The Article], 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 [51Jump To The Next Citation Point In The Article]). 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 tex2html_wrap_inline3015 grow with volume and may well reach the value 1 expected for a first-order transition as tex2html_wrap_inline3017 .

The origin of this behaviour was further elucidated by Bialas et al [49Jump To The Next Citation Point In The Article, 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 [77Jump To The Next Citation Point In The Article], who made a related mean-field ansatz, with the local weights depending on the local entropies around the vertices.



4.7 Influence of the measure4 Dynamical triangulations4.5 The phase structure

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
© Max-Planck-Gesellschaft. ISSN 1433-8351
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