Existence of an exponential bound?

4.3 Existence of an exponential bound?

As a consequence of identities and inequalities satisfied by the $Ni$ ([8

] contains a detailed discussion), the action (21

) is bounded above and below for fixed volume $N4$ . If the number of configurations for fixed $N 4$ , is exponentially bounded as $N → ∞ 4$ , that is, $Z(κ ,N ) 2 4$ grows at most as $constN4 Z (κ2,N4 ) ∼ e$ , there is a “critical line” $c κ4 = κ4(κ2)$ in the $(κ2,κ4 )$ -plane, where for fixed $κ2$ , $Z (κ2,κ4)$ converges for $c κ4 > κ4(κ2)$ . True critical behaviour may be found by approaching suitable points on this line from the region above the line, where $Z$ is well defined.

Doubts on the existence of an exponential bound were raised by Catterall et al. [74 ], who considered the behaviour of $Ω$ in $∑ −κ4N4 Z (κ0,κ4) = N4 e Ω (κ0,N4 )$ . Their data (taken for $N4 ≤ 32k$ ) were consistent with a leading factorial behaviour $Ω ∼ (N4!)δ$ . The same scenario was favoured by de Bakker and Smit [89 ], who performed further investigations of $κc4$ . Subsequently, Ambjørn and Jurkiewicz [12 ] and Brügmann and Marinari [60 ] added further data points at $N = 64k 4$ and $N = 128k 4$ respectively. Their numerical results, as well as those by Catterall et al. [78 ], who employed an alternative method for measuring $Ω$ , 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 ].