Probability Seminar: Zichun Ye

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.