Models of Gradient Type with Sub-Quadratic Actions

We consider models of gradient type, which are the densities of a collection of real-valued random variables $\varphi :=\{{\varphi }_x:x\in \mathrm{\Lambda }\}$ given by $Z^{-1}\exp (-{\sum }_{j\sim k}V({\varphi }_j-{\varphi }_k))$. We focus our study on the case that $V(\mathrm{\nabla }\varphi )=[1+(\mathrm{\nabla }\varphi {)}^2{]}^{\alpha }$ with $0 < \alpha < 1/2$,whichisanon-convexpotential\mathrm{.}Weintroduceanauxiliaryfield$t_{jk}$ for each edge and represent the model as the marginal of a model with log-concave density. Based on this method, we prove that finite moments of the fields $<[v\cdot \varphi {]}^p>$ are bounded uniformly in the volume. This leads to the existence of infinite volume measures. Also, every translation invariant, ergodic infinite volume Gibbs measure for the potential $V$ above scales to a Gaussian free field.