  QCD with infinite heavy quark masses exhibits a first-order thermal
transition which is driven by the spontaneous breaking of the global
$\mathcal{Z}_3$ center symmetry. We analyze the corresponding order parameter,
namely the Polyakov loop and its moments, and show, with a rigorous finite size
scaling, that in the continuum limit the transition is of first order. We show
that the use of a parallel tempering algorithm can significantly reduce the
large auto-correlation times which are mainly caused by the supercritical
slowing down. As a result, we calculate the transition temperature $w_0 T_c$
with per-mill precision, and the latent heat, carrying out controlled continuum
and infinite volume extrapolations.
