4.4 Performing the state sum4 Dynamical triangulations4.2 Path integral for dynamical

4.3 Existence of an exponential bound?

As a consequence of identities and inequalities satisfied by the tex2html_wrap_inline2901 ([8Jump To The Next Citation Point In The Article] contains a detailed discussion), the action (21Popup Equation) is bounded above and below for fixed volume tex2html_wrap_inline2863 . If the number of configurations for fixed tex2html_wrap_inline2863, is exponentially bounded as tex2html_wrap_inline2907, that is, tex2html_wrap_inline2873 grows at most as tex2html_wrap_inline2911, there is a ``critical line'' tex2html_wrap_inline2913 in the tex2html_wrap_inline2915 -plane, where for fixed tex2html_wrap_inline2917, tex2html_wrap_inline2919 converges for tex2html_wrap_inline2921 . 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 tex2html_wrap_inline2925 in tex2html_wrap_inline2927 . Their data (taken for tex2html_wrap_inline2929) were consistent with a leading factorial behaviour tex2html_wrap_inline2931 . The same scenario was favoured by de Bakker and Smit [89Jump To The Next Citation Point In The Article], who performed further investigations of tex2html_wrap_inline2933 . Subsequently, Ambjørn and Jurkiewicz [12] and Brügmann and Marinari [60] added further data points at tex2html_wrap_inline2935 and tex2html_wrap_inline2937 respectively. Their numerical results, as well as those by Catterall et al [78], who employed an alternative method for measuring tex2html_wrap_inline2925, 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 [8Jump To The Next Citation Point In The Article].



4.4 Performing the state sum4 Dynamical triangulations4.2 Path integral for dynamical

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
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