Delocalisation and absolute-value-FKG in the solid-on-solid model

• 成果类型: Article
• 署名作者: Lammers, Piet; Ott, Sebastien
• 署名单位: Universite Paris Saclay; University of Fribourg
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01202-y
• 发表日期: 2024
• 页码: 63-87
• 关键词: phase-transitions

摘要

The solid-on-solid model is a model of height functions, introduced to study the interface separating the + and - phase in the Ising model. The planar solid-on-solid model thus corresponds to the three-dimensional Ising model. Delocalisation of this model at high temperature and at zero slope was first derived by Frohlich and Spencer, in parallel to proving the Berezinskii-Kosterlitz-Thouless phase transition. The first main result of this article consists of a simple, alternative proof of delocalisation of the solid-on-solid model. In fact, the argument is more general: it works on any planar graph-not just the square lattice-and implies that the interface delocalises at any slope rather than exclusively at the zero slope. The second main result, proved independently, is that the absolute value of the height function in this model satisfies the FKG lattice condition. This property is believed to be intimately linked to the (quantitative) understanding of delocalisation, given the recent successes in the context of the square ice and (more generally) the six-vertex model, and it has already been used elsewhere in a new proof of the BKT transition. The new FKG inequality is shown to hold true for both the solid-on-solid model as well as for the discrete Gaussian model, which in this article implies that the two notions of delocalisation, namely delocalisation in finite volume and delocalisation of shift-invariant Gibbs measures, coincide.