First simulations

3.3 First simulations

The first numerical studies of the Regge action were undertaken by Berg [38 , 39

] and Hamber and Williams [118

, 119 ]. Berg performed a Monte Carlo simulation of the pure-curvature action for hypercubic 2⁴- and 3⁴-lattices with simplicial subdivision (see [40

] for a description of the method). He used the scale-invariant measure $∏ dli 𝒟l = i li$ . To avoid the divergence that results from rescaling the link lengths, he kept the total volume constant by performing an overall length rescaling of all links after each move. This amounts to fixing a typical length scale $l0 := (v0)14$ , where $v0$ is the expectation value of the 4-simplex volume.

For $k = 0$ , he found a negative average curvature $⟨R ⟩$ , and some evidence for a canonical scaling behaviour of lengths, areas and volumes. For $k = ±0.3$ , he obtained a negative (positive) average deficit angle $⟨δ⟩$ and a positive (negative) $⟨R⟩$ . A more detailed analysis for $0 ≤ k ≤ 0.1$ led Berg [39 ] to conclude that there exists a critical value $kc$ (presumably a first-order transition [40 ]), below which $⟨R ⟩$ is convergent, whereas above it diverges. (Myers [169 ] has conjectured that it may be possible to perform a similar analysis for Monte Carlo data for the Lorentzian action.)

By contrast, Hamber and Williams [118 ] simulated the higher-derivative action

on 2⁴- and 4⁴-lattices, using a time-discretized form of the Langevin evolution equation (see also [106 ]). For technical reasons, one uses barycentric instead of Voronoi volumes. They employed the scale-invariant measure $∏ 𝒟l = i dliF 𝜖(l) li$ , where $F𝜖$ enforces an ultra-violet cutoff $𝜖$ . They investigated the average curvature $ℛ ∼ ⟨R ⟩$ and squared curvature $2 ℛ$ (scaled by powers of $2 ⟨l ⟩$ to make them dimensionless), as well as $2 ⟨δb⟩$ and $2 ⟨Vb⟩∕⟨l ⟩$ . For $λ = k = a = 0$ , one finds a negative $ℛ$ and a large $2 ℛ$ , indicating a rough geometry. For small $a$ , one observes a sudden decrease in $ℛ2$ , as well as a jump from large positive to small negative values of $ℛ$ as $λ$ is increased. For large $a$ , $ℛ$ is small and negative, and the geometry appears to be smooth. Like Berg, they advocated a fundamental-length scenario, where the dynamically determined average link length provides an effective UV-cutoff.