[Paper Review] Delocalisation and absolute-value-FKG in the solid-on-solid model

Piet Lammers, Sébastien Ott|arXiv (Cornell University)|Jan 13, 2021

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper presents a new, general proof of delocalisation in the solid-on-solder (SOS) model on any planar graph, extending beyond zero slope to arbitrary inclinations. It further establishes that the absolute value of the height function satisfies the FKG lattice condition, which links to quantitative delocalisation and implies equivalence between finite-volume and infinite-volume delocalisation in both the SOS and discrete Gaussian models.

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 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 Fröhlich 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.

Motivation & Objective

• To provide a simpler, more general proof of delocalisation in the SOS model beyond zero slope and the square lattice.
• To establish that the absolute value of the height function satisfies the FKG lattice condition in the SOS model.
• To show that the FKG property implies equivalence between finite-volume delocalisation and delocalisation in shift-invariant Gibbs measures.
• To extend the applicability of FKG-based methods to the SOS and discrete Gaussian models, supporting quantitative understanding of delocalisation.
• To contribute to the rigorous analysis of interface fluctuations in effective interface models, particularly in three dimensions.

Proposed method

• Use of a constructive, probabilistic argument based on conditional measures and coupling with the random-cluster model to analyze the sign structure of the height function.
• Application of the Edwards-Sokal coupling to relate the sign process to a percolation model with increasing interaction strengths.
• Leveraging monotonicity and FKG inequality for the joint law of |φ| and the percolation configuration ω in finite volumes.
• Proving convergence of finite-volume measures μΔₙ to a unique ergodic Gibbs measure μ via local convergence and percolation arguments.
• Using Burton and Keane’s result to show that ω has at most one infinite cluster, and ruling out infinite clusters to ensure uniqueness of μ.
• Establishing that the FKG property for |φ| holds in both the SOS and discrete Gaussian models through joint monotonicity and coupling techniques.

Experimental results

Research questions

• RQ1 Does delocalisation in the SOS model hold for arbitrary slopes, not just zero slope, on general planar graphs?
• RQ2 Does the absolute value of the height function in the SOS model satisfy the FKG lattice condition?
• RQ3 Can the FKG property for |φ| be used to establish equivalence between finite-volume delocalisation and delocalisation in shift-invariant Gibbs measures?
• RQ4 Is the FKG inequality for |φ| also valid in the discrete Gaussian model, and what are its implications?
• RQ5 Can the FKG property help in obtaining quantitative bounds on interface fluctuations in effective interface models?

Key findings

• The paper establishes a new, general proof of delocalisation for the SOS model on any planar graph, valid for all slopes, not just zero slope.
• The absolute value of the height function |φ| satisfies the FKG lattice condition in the SOS model, a novel result with deep implications.
• The FKG property for |φ| implies that finite-volume delocalisation and delocalisation in shift-invariant Gibbs measures coincide in the SOS model.
• The same FKG property holds for the discrete Gaussian model, leading to the same equivalence between finite-volume and infinite-volume delocalisation.
• The unique ergodic Gibbs measure μ with μ(φᵣ) = 0 is the weak limit of finite-volume measures μΔₙ as Δₙ exhausts the lattice, under mild conditions on the sequence (Δₙ).
• The joint law of (|φ|, ω) is increasing in the volume Λ, and the percolation process ω does not contain an infinite cluster almost surely, ensuring uniqueness of μ.