  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 [3,4]. However, one may wonder if this
finding was affected by the numerical methods used: to control volume
fluctuations, in both studies [3,4] an artificial harmonic potential was added
to the action; in [4] 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 [6]. With these improved methods, on
systems of size up to 64k 4-simplices, we confirm that the phase transition is
first order.
