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The transition between the two phases of 4D Euclidean Dynamical Triangulation[1] was long believed to be of second order until in 1996 first order behaviorwas found for sufficiently large systems [5,9]. However, one may wonder if thisfinding was affected by the numerical methods used: to control volumefluctuations, in both studies [5,9] an artificial harmonic potential was addedto the action; in [9] measurements were taken after a fixed number of acceptedinstead of attempted moves which introduces an additional error. Finally thesimulations suffer from strong critical slowing down which may have beenunderestimated. In the present work, we address the above weaknesses: we allowthe volume to fluctuate freely within a fixed interval; we take measurementsafter a fixed number of attempted moves; and we overcome critical slowing downby using an optimized parallel tempering algorithm [12]. With these improvedmethods, on systems of size up to 64k 4-simplices, we confirm that the phasetransition is first order.  In addition, we discuss a local criterion to decide whether parts of atriangulation are in the elongated or crumpled state and describe a newcorrespondence between EDT and the balls in boxes model. The latter gives riseto a modified partition function with an additional, third coupling. Finally,we propose and motivate a class of modified path-integral measures that mightremove the metastability of the Markov chain and turn the phase transition intosecond order.

Euclidean Dynamical Triangulation revisited: is the phase transition really 1st order? (extended version)