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 ϕ ≔ {ϕx: x ∈ Λ} given by Z−1 exp(−∑j∼kV(ϕj − ϕk)). We focus our study on the case that V(∇ϕ)=[1+(∇ϕ)2]α with 0 < α < 1/2, which is a nonconvex potential. We introduce an auxiliary field tjk 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⋅ϕ]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.

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