The transition between the two phases of 4D Euclidean Dynamical Triangulation  was long believed to be of second order until in 1996 first order behavior was found for sufficiently large systems [5, 9]. However, one may wonder if this finding was affected by the numerical methods used: to control volume fluctuations, in both studies [5, 9] an artificial harmonic potential was added to the action and in  measurements were taken after a fixed number of accepted instead of attempted moves which introduces an additional error. Finally the simulations suffer from strong critical slowing down which may have been underestimated.
In the present work, we address the above weaknesses: we allow the volume to fluctuate freely within a fixed interval; we take measurements after a fixed number of attempted moves; and we overcome critical slowing down by using an optimized parallel tempering algorithm . With these improved methods, on systems of size up to N4 = 64k 4- simplices, we confirm that the phase transition is 1st order.
In addition, we discuss a local criterion to decide whether parts of a triangulation are in the elongated or crumpled state and describe a new correspondence between EDT and the balls in boxes model. The latter gives rise to a modified partition function with an additional, third coupling.
Finally, we propose and motivate a class of modified path-integral measures that might remove the metastability of the Markov chain and turn the phase transition into 2nd order.
Models of Quantum Gravity Lattice Models of Gravity
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
H.G. Katzgraber, S. Trebst, D.A. Huse and M. Troyer, Feedback-optimized parallel tempering Monte Carlo, J. Stat. Mech. (2006) P03018 [cond-mat/0602085].
B. Bauer, E. Gull, S. Trebst, M. Troyer and D.A. Huse, Optimized broad-histogram simulations for strong first-order phase transitions: Droplet transitions in the large-Q Potts model, J. Stat. Mech. (2010) P01020 [arXiv:0912.1192].
A.M. Ferrenberg and R.H. Swendsen, New Monte Carlo technique for studying phase transitions, Erratum Phys. Rev. Lett.63 (1989) 1658.ADSCrossRefGoogle Scholar
K. Binder, K. Vollmayr, H.-P. Deutsch, J.D. Reger, M. Scheucher and D.P. Landau, Monte Carlo Methods for First Order Phase Transitions: Some Recent Progress, Int. J. Mod. Phys.C 3 (1992) 1025.ADSCrossRefGoogle Scholar
U. Pachner, P.L. homeomorphic manifolds are equivalent by elementary 5hellingst, Eur. J. Combinator.12 (1991) 129.CrossRefMATHGoogle Scholar