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UBC Theses and Dissertations

Models of gradient type with sub-quadratic actions and their scaling limits Ye, Zichun


The main results of this thesis concern models of gradient type with sub-quadratic actions and their scaling limits. The model of gradient type is the density of a collection of real-valued random variables Φ := {Φx : x ∈ Λ } given by ℤ-¹ exp(-∑ j∼k V(Φj-Φk)) the Gibbs measure with nearest neighbor interaction and the potential function V. We focus our study on the case that V(∇Φ) = [1+(∇Φ)²]^α with 0 < α < 1/2, which is a non-convex potential. The first result concerns the thermodynamic limits of the model of gradient type. We introduce an auxiliary field 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 for the finite volume measure. This bound leads to the existence of infinite volume measures. The second result is the random walk representation and the scaling limit of the translation-invariant, ergodic gradient infinite volume Gibbs measure. We represent every infinite volume Gibbs measure as a mixture over Gaussian gradient measures with a random coupling constant w xy for each edge. With such representation, we give an estimation on the decay of the two point correlation function. Then by the quenched functional central limit theorem in random conductance model, we prove that every ergodic, infinite volume Gibbs measure with mean zero for the potential V above scales to a Gaussian free field.

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