Performing the state sum

4.4 Performing the state sum

The partition function is evaluated numerically with the help of a Monte Carlo algorithm (see [57

, 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 $S4$ 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 [79 , 51 ], but the lattice sizes and fluctuations may still be too small to detect a potential failure of ergodicity, cf. [53 ].

To improve the efficiency of the algorithm, Ambjørn and Jurkiewicz [14 ] 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 $c γ (κ 2) ≈ 0$ [10 , 14 , 97 ]. More recently, Egawa et al. [98 ] have reported a value of $γ ≈ 0.26$ .