Models of Gradient Type with Sub-Quadratic Actions

TL;DR

This paper studies non-convex gradient models with a specific potential, introducing an auxiliary field method to prove bounded moments and the existence of infinite volume measures, showing these measures scale to a Gaussian free field.

Contribution

It introduces an auxiliary field approach for non-convex gradient models, establishing bounded moments and the existence of infinite volume measures, and demonstrates scaling to a Gaussian free field.

Findings

Bounded finite moments of the fields are established.

Existence of infinite volume Gibbs measures is proven.

Scaling limits of measures are Gaussian free fields.

Abstract

We consider models of gradient type, which are the densities 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-concave 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 translation invariant, ergodic infinite volume Gibbs measure for the potential $V$ above scales to a Gaussian free field.