Models of Gradient Type with Sub-Quadratic Actions

Abstract

We consider models of gradient type, which are the densities of a collection of real-valued random variables $ϕ:=\{ϕ_x: x \in Λ\}$ given by $Z^{-1}\exp({-\sum\nolimits_{j \sim k}V(ϕ_j-ϕ_k)})$. We focus our study on the case that $V(\nablaϕ) = [1+(\nablaϕ)^2]^α$ with $0 < α< 1/2$, 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-concave density. Based on this method, we prove that finite moments of the fields $\left<[v \cdot ϕ]^p \right>$ 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.