LARGE-FIELD VERSUS SMALL-FIELD EXPANSIONS AND SOBOLEV INEQUALITIES

We study a model for a two-dimensional random interface phi(x), x is an element of R(2), described by a massless Gaussian measure perturbed by a weak potential V(phi)=(epsilon(2)/2)(e(-alpha phi) - 1)(2). Such a model occurs, for instance, in a phenomenological description of the wetting transition. We prove that, provided alpha is small enough, the two-point function decreases exponentially with a rate of order m=epsilon alpha, which is just the mean-field value. The large-field-region problem due to the fact that V(phi) remains bounded when phi --> +infinity is treated by means of a large-field versus small-field expansion combined with elementary Sobolev inequalities. The paper is intended to be accessible to nonexperts.