One dimensional lattice gases with rapidly decreasing interaction

We consider the pressure and the correlation functions of a one dimensional lattice gas in which the mutual interaction decreases as r exp-n^t, (r, t>0), when the interparticle distance n. We prove that such a system cannot show phase transitions of order k1 in the sense that the pressure and the correlation functions are infinitely differentiable with respect to any relevant parameter (such as the temperature or the chemical potential).