Smooth phase in the one-dimensional discrete Gaussian model with 1/(?j)2 interaction at inverse temperature ?>1

Abstract

We consider the one-dimensional discrete Gaussian model with interaction energyg satisfyingg(i, j)=g(i−j)∼1/(i− j)2 and prove that for the inverse temperatureβ>1 this system displays a smooth phase characterized by ⩽C<∞ if the nearest neighbor coupling g(1) is sufficiently large. Our method also allows us to treat the 1/(i− j)2 Ising model and reproves the existence of spontaneous magnetization under the above conditions.