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 Point5072] 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 [171172105] (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 S4 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-N4 barriers, which should manifest themselves as an obstacle to cooling down an large initial random triangulation to the minimal S4-configuration. No such barriers were found for system sizes ≤ 64k, but unfortunately they were also absent for an analogous simulation (for N5 ≤ 32k) performed by de Bakker [86] for 5 S, 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 Z (κ2,κ4), but adds a potential term to the action so that the only relevant contributions come from states in an interval [V − ΔN4, V + ΔN4 ] 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 Point51Jump To The Next Citation Point], but the lattice sizes and fluctuations may still be too small to detect a potential failure of ergodicity, cf. [53Jump To The Next Citation Point].

To improve the efficiency of the algorithm, Ambjørn and Jurkiewicz [14Jump To The Next Citation Point] 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 γ, assuming a behaviour of the form γ(κ2)−3 κc4(κ2)N4 Z(κ2,N4 ) = N 4 e × (1 + O (1∕N4 )), by counting baby universes of various sizes. At the transition point, one finds c γ (κ 2) ≈ 0 [1014Jump To The Next Citation Point97]. More recently, Egawa et al. [98] have reported a value of γ ≈ 0.26.


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