Models of Gradient Type with Sub-Quadratic Action

We consider models of gradient type, which is the density of a collection of real-valued random variables \(\phi :=\{\phi_x: x \in \Lambda\}\) given by \(Z^{-1}\exp({-\sum\nolimits_{j \sim k}V(\phi_j-\phi_k)})\). We focus our study on the case that \(V(\nabla\phi) = [1+(\nabla\phi)^2]^\alpha\) with \(0 < \alpha < 1/2\), which is a non-convex potential. We introduce an auxiliary field \(t_{jk}\) for each edge and represent the model as the marginal of a model with log-cancave density. Based on this method, we prove that finite moments of the fields \(\left<[v \cdot \phi]^p \right>\) are bounded uniformly in the volume. This leads to the existence of infinite volume measures. Also, every ergodic infinite volume Gibbs measure with mean zero for the potential \(V\) above scales to a Gaussian free field.