Confinement, Deconfinement and Freezing in Lattice Yang-Mills Theories with Continuous Time

In this paper I analyse lattice Yang-Mills theories with continuous time. After a short discussion of more conceptual questions, such as the existence of a Hamilton operator in the infinite volume limit, I study the phase diagram. The existence of a strong coupling/low temperature confinement phase (which was not proven up to now) is established for arbitrary compact groups, continuous or discrete. For discrete compact groups the deconfinement region decomposes into (at least) two phases, which are distinguished by the behaviour of spatial Wilson loops: a deconfinement phase where spatial Wilson loops still show area law behaviour, and a “freezing” phase with perimeter law behaviour for spatial Wilson loops. The methods to prove these results rely on cluster expansion methods, combined with renormalisation ideas.