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

Abstract

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 solidon-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 Fröhlich and Spencer, in parallel to proving the Kosterlitz-Thouless phase transition. The first main result of this article consists of a simple, alternative proof for 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 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. 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 implies immediately that the two notions of delocalisation, namely delocalisation in finite volume and delocalisation of shift-invariant Gibbs measures, coincide.

An Elementary Proof of Phase Transition in the Planar XY Model

“…In this section, we prove that the height function satisfies the absolute-value-FKG property, which is known to imply the dichotomy in Theorem 4 below [8,20]. Here we will only work with the potential V β as defined in (4).…”

“…This is a consequence of the absolute-value-FKG property (Proposition 3) and standard arguments using monotonicity in boundary conditions. See [20,Theorem 2.7].…”

The Discrete Gaussian model, I. Renormalisation group flow at high temperature

“…The rigorous approach of [42] (see also [43]) uses a multiscale resummation based on conditional expectations and Jensen's inequality. For a recent exposition as well as recent extensions and applications of this approach, see [46,47,55,67], and for recent alternative approaches to the proof of the existence of the Kosterlitz-Thouless transition, see also [2,58,59,66]. These approaches have many appealing features which include that they apply quite robustly to various models, but they are not precise enough to derive scaling limits or sharp asymptotics of correlation functions, or to study the (expected) critical curve -the Kosterlitz-Thouless transition line, see Fig.…”

Weakly non-planar dimers

“…In a broader perspective, there has been a number of recent results (e.g. [1,12,36]) that prove delocalization of discrete, two-dimensional interface models at high temperature, even though they fall short of proving convergence to the GFF. Let us mention in particular the recent [36], which proves with a rather soft argument a (non-quantitative) delocalization statement for rather general height models, under the restriction, however, that the underlying graph has maximal degree three.…”

“…[1,12,36]) that prove delocalization of discrete, two-dimensional interface models at high temperature, even though they fall short of proving convergence to the GFF. Let us mention in particular the recent [36], which proves with a rather soft argument a (non-quantitative) delocalization statement for rather general height models, under the restriction, however, that the underlying graph has maximal degree three. For the particular case of the 6-vertex model, delocalization of the height function is known to hold in several regions of parameters [14,15,38,42] but full scaling to the GFF has been proven only in a neighborhood of the free fermion point [27].…”