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On the duality between height functions and continuous spin models

Diederik van Engelenburg, Marcin Lis

TL;DR

This work develops a comprehensive duality framework between integer-valued height functions with positive definite potentials and circle-valued spin models, translating observables across the dual pair. Central results include a universal GFF-type bound on height-variance via the Green’s function, localisation on transient graphs, monotonicity of height-variance with inverse temperature, and precise links between height delocalisation and BKT-type behavior in two dimensions, with an emphasis on XY-like models. The authors also establish that certain spin-observables project to GFF-like covariance, prove a planar central limit theorem for gradient observables, and provide a concise duality-based route to delocalisation implications for the spin system. Methodologically, the paper develops covariance duality, extends Ginibre-type inequalities to these dual models, and employs graph-modification and convexity reduction techniques to derive sharp phase-transition statements in planar and nearly planar geometries. Together, these results illuminate the deep interplay between height-function delocalisation, spin correlations, and phase transitions across a broad class of abelian spin and height-function models.

Abstract

We revisit the classical phenomenon of duality between random integer-valued height functions with positive definite potentials and abelian spin models with O(2) symmetry. We use it to derive new results in quite high generality including: a universal upper bound on the variance of the height function in terms of the Green's function (a GFF bound) which among others implies localisation on transient graphs; monotonicity of said variance with respect to a natural temperature parameter; the fact that delocalisation of the height function implies a BKT phase transition in planar models; and also delocalisation itself for height functions on periodic ``almost'' planar graphs.

On the duality between height functions and continuous spin models

TL;DR

This work develops a comprehensive duality framework between integer-valued height functions with positive definite potentials and circle-valued spin models, translating observables across the dual pair. Central results include a universal GFF-type bound on height-variance via the Green’s function, localisation on transient graphs, monotonicity of height-variance with inverse temperature, and precise links between height delocalisation and BKT-type behavior in two dimensions, with an emphasis on XY-like models. The authors also establish that certain spin-observables project to GFF-like covariance, prove a planar central limit theorem for gradient observables, and provide a concise duality-based route to delocalisation implications for the spin system. Methodologically, the paper develops covariance duality, extends Ginibre-type inequalities to these dual models, and employs graph-modification and convexity reduction techniques to derive sharp phase-transition statements in planar and nearly planar geometries. Together, these results illuminate the deep interplay between height-function delocalisation, spin correlations, and phase transitions across a broad class of abelian spin and height-function models.

Abstract

We revisit the classical phenomenon of duality between random integer-valued height functions with positive definite potentials and abelian spin models with O(2) symmetry. We use it to derive new results in quite high generality including: a universal upper bound on the variance of the height function in terms of the Green's function (a GFF bound) which among others implies localisation on transient graphs; monotonicity of said variance with respect to a natural temperature parameter; the fact that delocalisation of the height function implies a BKT phase transition in planar models; and also delocalisation itself for height functions on periodic ``almost'' planar graphs.
Paper Structure (25 sections, 28 theorems, 108 equations, 2 figures)

This paper contains 25 sections, 28 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General duality
  3. Discrete calculus
  4. Spin models and random height functions
  5. Upper bound on the variance of the height function
  6. Delocalisation implies the BKT phase transition in two dimensions
  7. Background
  8. Notions of delocalisation
  9. Setup
  10. The implication
  11. The case of the $XY$ model
  12. An equivalence
  13. Monotonicity of variance of the height function in temperature
  14. Setup
  15. Increasing variance
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $\# \in \{\Diamond, \star\}$ and let $-\#$ denote the other element of $\{\Diamond, \star\}$. Then for any $\epsilon \in \Omega^1(\mathbb{R})$, we have

Figures (2)

  • Figure 1: Left: original graph with $v_0$ in the center. Middle two: first step of the degree-reduction algorithm, dotted lines correspond to vertices to be glued together. Right: Final graph after "star--tree" transform, with vertices glued together and all vertices have three or less neighbors.
  • Figure 2: Left: an example of $\mathbb{Z}^2$ with long-range interactions; only the edges of the (red) origin are drawn. Middle: gluing. The square (gray) vertices are added, together with the green edges where the gluing will happen. Right: the final (planar) graph.

Theorems & Definitions (59)

  • Definition 1: Height function and spin potentials
  • Example 1
  • Lemma 2.1: Fourier--Pontryagin duality
  • Remark 1
  • Remark 2
  • Lemma 2.2: Covariance duality
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 49 more