  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 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; in [9] 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 [12]. With these improved
methods, on systems of size up to 64k 4-simplices, we confirm that the phase
transition is first 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
second order.
