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

TL;DR

This paper extends the renormalisation group analysis of gradient models with non-convex potentials to infinite volume, proving strict convexity of the free energy and the Gaussian free field scaling limit without subsequence extraction.

Contribution

It advances previous work by removing the subsequence requirement in the analysis of surface tension and scaling limits for non-convex gradient models.

Findings

Proved strict convexity of free energy in the infinite-volume limit.

Established the Gaussian free field as the scaling limit without subsequences.

Extended renormalisation group methods to infinite volume.

Abstract

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].