Gluon propagator close to criticality

There are good reasons that the deconfinement phase transition of pure Yang-Mills theory at finite temperature should also be reflected in the behavior of gauge-fixed gluonic correlation functions. Understanding this in detail would provide another important example of how physical information can be extracted from gauge-dependent correlations, which is not always obvious. Therefore, herein 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.