Avoiding collapse

3.7 Avoiding collapse

Starting from the observation that Regge configurations exist with $Rtot → ∞$ and $Vtot → 0$ , Beirl et al. [31

, 32 ] investigated the influence of a cutoff $f$ on the fatness of a simplex $σ$ , defined by

Such a uniform shrinking of simplices is known to be necessary in order for piecewise flat manifolds to approach their continuum counterparts [82 ]. On the 4⁴-lattice, with the $ldl$ -measure and $λ = 0$ , they determined $ℛ (k)$ for decreasing $f$ . For small $k$ , the choice of $f$ seems to have only little influence, but towards the transition point $kc$ , $⟨ϕσ⟩$ simultaneously decreases. Next they investigated a variety of measures of the form $2σ−1 l dl$ (see also [35 ]). For $σ ≤ 1$ , only a mild $σ$ -dependence of $ℛ$ is observed as $k → kc$ . However, for $σ = 1.5$ , there are significant differences for the entire range of $k$ , and some evidence that the geometry degenerates.