For , he found a negative average curvature , and some evidence for a canonical scaling behaviour of lengths, areas and volumes. For , he obtained a negative (positive) average deficit angle and a positive (negative) . A more detailed analysis for led Berg  to conclude that there exists a critical value (presumably a first-order transition ), below which is convergent, whereas above it diverges. (Myers  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  simulated the higher-derivative action4- and 44-lattices, using a time-discretized form of the Langevin evolution equation (see also ). For technical reasons, one uses barycentric instead of Voronoi volumes. They employed the scale-invariant measure , where enforces an ultra-violet cutoff . They investigated the average curvature and squared curvature (scaled by powers of to make them dimensionless), as well as and . For , one finds a negative and a large , indicating a rough geometry. For small , one observes a sudden decrease in , as well as a jump from large positive to small negative values of as is increased. For large , 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.
© Max Planck Society and the author(s)