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

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