  The deconfinement phase transition of pure Yang-Mills theory at finite
temperature is reflected in the behavior of gauge-fixed gluonic correlation
functions. This is one of many examples of how physical information can be
extracted from gauge-dependent correlations which is not always obvious. Here,
we study the behavior of the Landau-gauge gluon propagator of pure SU(2) across
the phase transition in 2+1 and 3+1 dimensions in order to assess to what
extend the corresponding critical behavior is reflected in these correlations.
We discuss why it should emerge from a continuum perspective and test our
expectations in lattice simulations. A comparison with SU(3) furthermore
reveals quite clear indications for a sensitivity of the gluon propagator to
the order of the transition.
