### 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 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- 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 [86] 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. [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 , 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. [98] have reported a value
of .