Scaling limit and strict convexity of free energy for gradient models with non-convex potential

20 Jul 2020  ·  Hilger Susanne ·

We consider gradient models on the lattice $\mathbb{Z}^d$. These models serve as effective models for interfaces and are also known as continuous Ising models. The height of the interface is modelled by a random field with an energy which is a non-convex perturbation of the quadratic interaction. We are interested in the Gibbs measure with tilted boundary condition $u$ at inverse temperature $\beta$ of this model. In [AKM16], [Hil16] and [ABKM19] the authors show that for small tilt $u$ and large inverse temperature $\beta$ the surface tension is strictly convex, where the limit is taken on a subsequence. Moreover, it is shown that the scaling limit (again on a subsequence) is the Gaussian free field on the continuum torus. The method of the proof is a rigorous implementation of the renormalisation group method following a general strategy developed by Brydges and coworkers. In this paper the renormalisation group analysis is extended from the finite-volume flow to an infinite-volume version to eliminate the necessity of the subsequence in the results in [AKM16], [Hil16] and [ABKM19].

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