4.5 The phase structure4 Dynamical triangulations4.3 Existence of an exponential

4.4 Performing the state sum

The partition function is evaluated numerically with the help of a Monte-Carlo algorithm (see [57Jump To The Next Citation Point In The Article, 50, 72] for details). There is a set of five topology-preserving moves which change a triangulation locally, and which are ergodic in the grand canonical ensemble [171, 172, 105] (see also [8] for a discussion). No ergodic finite set of topology- and volume-preserving moves exists for generic four-dimensional manifolds. This prevents one from using the canonical (i.e. volume-preserving) ensemble. If tex2html_wrap_inline2837 is not algorithmically recognizable in the class of all piecewise linear (or smooth) 4d manifolds, the numerical simulations may miss out a substantial part of the state space because of the absence of ``computational ergodicity'' [170].

An attempt was made by Ambjørn and Jurkiewicz [13] to link the non-recognizability to the presence of large- tex2html_wrap_inline2863 barriers, which should manifest themselves as an obstacle to cooling down an large initial random triangulation to the minimal tex2html_wrap_inline2837 -configuration. No such barriers were found for system sizes tex2html_wrap_inline2947, but unfortunately they were also absent for an analogous simulation (for tex2html_wrap_inline2949) performed by de Bakker [86] for tex2html_wrap_inline2951, which is not recognizable.

Since the local moves alter the volume, one works in practice with a ``quasi-canonical'' ensemble, i.e. one uses the grand canonical ensemble tex2html_wrap_inline2919, but adds a potential term to the action so that the only relevant contributions come from states in an interval tex2html_wrap_inline2955 around the target volume V . There have been several cross-checks which have found no dependence of the results on the width and shape of the potential term [79Jump To The Next Citation Point In The Article, 51Jump To The Next Citation Point In The Article], but the lattice sizes and fluctuations may still be too small to detect a potential failure of ergodicity, c.f. [53Jump To The Next Citation Point In The Article].

To improve the efficiency of the algorithm, Ambjørn and Jurkiewicz [14Jump To The Next Citation Point In The Article] used additional global (topology-preserving) ``baby universe surgery'' moves, by cutting and gluing pieces of the simplicial complex . In the branched polymer phase, one can estimate the entropy exponent tex2html_wrap_inline2959, assuming a behaviour of the form tex2html_wrap_inline2961, by counting baby universes of various sizes. At the transition point, one finds tex2html_wrap_inline2963 [10, 14Jump To The Next Citation Point In The Article, 97]. More recently, Egawa et al [98] have reported a value of tex2html_wrap_inline2965 .



4.5 The phase structure4 Dynamical triangulations4.3 Existence of an exponential

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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