An attempt was made by Ambjørn and Jurkiewicz  to link the non-recognizability to the presence of large- barriers, which should manifest themselves as an obstacle to cooling down an large initial random triangulation to the minimal -configuration. No such barriers were found for system sizes , but unfortunately they were also absent for an analogous simulation (for ) performed by de Bakker  for , 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 , but adds a potential term to the action so that the only relevant contributions come from states in an interval around the target volume . There have been several cross-checks which have found no dependence of the results on the width and shape of the potential term [79, 51], but the lattice sizes and fluctuations may still be too small to detect a potential failure of ergodicity, cf. .
To improve the efficiency of the algorithm, Ambjørn and Jurkiewicz  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 , assuming a behaviour of the form , by counting baby universes of various sizes. At the transition point, one finds [10, 14, 97]. More recently, Egawa et al.  have reported a value of .
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