Theq-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops

Theq-state Potts model (both scalar and gauge versions) is rewritten, with the help of the duality transformation, into a form of the Pirogov-Sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough. This is then used in a new proof of the existence of a unique transition (inverse) temperature_t, where the mean internal energy is discontinuous. Moreover, we prove for the scalar model (again forq large enough) that there are discontinuities at_t of the magnetization and of the mass gap, with the magnetization vanishing below_t and the mass gap vanishing above_t. We also show that the surface tensions between ordered stable phases are strictly positive up to_t, and the surface tension between an ordered phase and the disordered one is strictly positive at_t. For the three-dimensional gauge model, the Wilson parameter exhibits a direct transition from an area law decay (quark confinement) to a perimeter law decay (deconfinement).On leave from ENS Rabat, Morocco.