Discrete field variables,the Coulomb gas,and low temperature behaviour

Field theories with discrete field variables are reformulated as continuous field theories coupled to a Coulomb gas. TheU(1) gauge theory in three dimensions and thexy-model possess dual versions that are treated this way. Low lying states of the Coulomb gas determine their behaviour at low temperature. The Coulomb vacuum gives rise to a gauge invariant low temperature expansion in powers of 1/β. After clarifying the meaning of exponential terms, we try to determine exponential contributions to Wilson's loop correlation function. These investigations are based on a simplified model of a generalized Villain type.