3.7 Avoiding collapse

Starting from the observation that Regge configurations exist with Rtot → ∞ and Vtot → 0, Beirl et al. [31Jump To The Next Citation Point32] investigated the influence of a cutoff f on the fatness of a simplex σ, defined by
---Vσ2--- ϕσ := maxl ∈σ l8 ≥ f > 0. (18 )

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.


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