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.