In this discussion @Seth Whitsitt mentioned in a comment that for gauge theories in statistical mechanics the Wilson lines are extremely important objects for the thermodynamic and phase properties of the considered system.
The concept of Wilson lines is roughly familar to me from QFT; at least it's formal definition and intiutive interpretation as parallel transport of the matter field, but haven't any really deep insight into this topic. Especially I'm not sure for what it is good for in QFT.
Q: Could somebody elaborate Seth Whitsitt's comment in more detail. "How much" do Wilson loops know about the system one is going to analyse? Precisely which important informations about the considered system (from viewpoint of statistical mechanics) the Wilson loops provide?
Can rougly the amount of information about the system deducable purely from analysis of Wilson loops on it be nearly compared with informations one can extract about the system from classical partition function $Z$ in classical statistical mechanics?
Motivation: maybe understanding better the role of Wilson loops in statistucal machanics provide a better insight in their importance for QFT. So the question is really focussed on setting statistical mechanics from only.
