Monte Carlo study of the Coulomb gas and the VillainXYmodel in the discrete Gaussian roughening representation

Monte Carlo simulations of the two-dimensional discrete Gaussian model of roughening are used to obtain the thermodynamic properties of the Coulomb lattice gas and the Villain $\mathrm{XY}$ model. A comparison of the results with the first four terms in the low-temperature expansion of the discrete Gaussian model and the independent-pair approximation in the Coulomb gas representation confirms the accuracy and reliability of the Monte Carlo method generally, but reveals difficulties in the neighborhood of the transition. Various types of Monte Carlo simulations are compared and their consequences in different representations are discussed.