### 4.3 Existence of an exponential bound?

As a consequence of identities and inequalities satisfied by the ([8] contains a detailed discussion),
the action (21) is bounded above and below for fixed volume . If the number of configurations
for fixed , is exponentially bounded as , that is, grows at most as
, there is a “critical line” in the -plane, where
for fixed , converges for . True critical behaviour may be found
by approaching suitable points on this line from the region above the line, where is well
defined.
Doubts on the existence of an exponential bound were raised by Catterall et al. [74], who considered the
behaviour of in . Their data (taken for ) were consistent
with a leading factorial behaviour . The same scenario was favoured by de Bakker and
Smit [89], who performed further investigations of . Subsequently, Ambjørn and Jurkiewicz [12] and
Brügmann and Marinari [60] added further data points at and respectively.
Their numerical results, as well as those by Catterall et al. [78], who employed an alternative method for
measuring , favour the existence of an exponential bound, although they cannot claim to be
conclusive.

There have also been theoretical arguments for the existence of an exponential bound, based on the
proofs of such bounds for the counting of minimal geodesic ball coverings of Riemannian spaces of bounded
geometry [68, 26], and the counting of discrete curvature assignments to unordered sets of
bones [8].