A theorem on the absence of phase transitions in one-dimensional growth models with on-site periodic potentials

We rigorously prove that a wide class of one-dimensional growth models with on-site periodic potential, such as the discrete sine-Gordon model, have no phase transition at any temperature T > 0. The proof relies on the spectral analysis of the transfer operator associated with the models. We show that this operator is Hilbert–Schmidt and that its maximum eigenvalue is an analytical function of temperature.