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The Ising magnetisation field and the Gaussian free field

Tomás Alcalde López, Lorca Heeney, Marcin Lis

TL;DR

This work builds a natural, probabilistic bosonisation-like link between the continuum Gaussian free field and the critical Ising magnetisation field in planar domains, showing four IMF copies are deterministically obtainable from a single GFF plus independent coin flips. The core technique blends a discrete Edwards–Sokal-type coupling with a novel double random current percolation model, and passes to the continuum via a multi-scale, $L^2$-type box-counting approximation that yields IMF decompositions into cluster-area measures of two-valued sets of the GFF, with the CLE$_4$ carpet emerging as the geometric backbone. The resulting continuum representation identifies the IMF as measurable functionals of the GFF, and provides a conformally covariant framework (via CLE$_4$-carpet measures) for the IMF boundaries and their interactions. The paper also outlines conjectural extensions to the Ashkin–Teller model, where the magnetisation/polarisation fields are tied to height-gap parameters and imaginary Gaussian multiplicative chaos, suggesting a broad, geometric realisation of bosonisation across a critical line and a path toward a full AT magnetisation theory in the continuum.

Abstract

We construct a natural coupling between the continuum Gaussian free field (GFF) and the critical Ising magnetisation field (IMF) in a planar domain. In fact, we show that two independent IMFs with $+$ boundary conditions and two independent IMFs with free boundary conditions are a deterministic function of a single instance of the GFF together with a sequence of independent coin flips. This construction should be seen as an extension of the bosonisation phenomenon, and to the best of our knowledge its existence has not been predicted before. We arrive at our main result in the continuum by studying novel discrete structures. Our starting point is a coupling resembling the Edwards-Sokal coupling between the Ising model and the Fortuin-Kasteleyn random cluster model, though with role of the latter played by a different percolation model obtained from the double random current model. By taking a scaling limit of the coupling at criticality, we obtain a continuum Edwards-Sokal-like representation of the IMFs in terms of certain two-valued sets of the GFF introduced by Aru, Sepúlveda and Werner.

The Ising magnetisation field and the Gaussian free field

TL;DR

This work builds a natural, probabilistic bosonisation-like link between the continuum Gaussian free field and the critical Ising magnetisation field in planar domains, showing four IMF copies are deterministically obtainable from a single GFF plus independent coin flips. The core technique blends a discrete Edwards–Sokal-type coupling with a novel double random current percolation model, and passes to the continuum via a multi-scale, -type box-counting approximation that yields IMF decompositions into cluster-area measures of two-valued sets of the GFF, with the CLE carpet emerging as the geometric backbone. The resulting continuum representation identifies the IMF as measurable functionals of the GFF, and provides a conformally covariant framework (via CLE-carpet measures) for the IMF boundaries and their interactions. The paper also outlines conjectural extensions to the Ashkin–Teller model, where the magnetisation/polarisation fields are tied to height-gap parameters and imaginary Gaussian multiplicative chaos, suggesting a broad, geometric realisation of bosonisation across a critical line and a path toward a full AT magnetisation theory in the continuum.

Abstract

We construct a natural coupling between the continuum Gaussian free field (GFF) and the critical Ising magnetisation field (IMF) in a planar domain. In fact, we show that two independent IMFs with boundary conditions and two independent IMFs with free boundary conditions are a deterministic function of a single instance of the GFF together with a sequence of independent coin flips. This construction should be seen as an extension of the bosonisation phenomenon, and to the best of our knowledge its existence has not been predicted before. We arrive at our main result in the continuum by studying novel discrete structures. Our starting point is a coupling resembling the Edwards-Sokal coupling between the Ising model and the Fortuin-Kasteleyn random cluster model, though with role of the latter played by a different percolation model obtained from the double random current model. By taking a scaling limit of the coupling at criticality, we obtain a continuum Edwards-Sokal-like representation of the IMFs in terms of certain two-valued sets of the GFF introduced by Aru, Sepúlveda and Werner.
Paper Structure (28 sections, 39 theorems, 243 equations, 8 figures)

This paper contains 28 sections, 39 theorems, 243 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. $\mathbf{(+)}$ For $+$ boundary conditions (Figure \ref{['fig:decomp-plus']}):
  4. $(\text{free})$ For free boundary conditions (Figure \ref{['fig:decomp-free']}):
  5. Further discussion and context
  6. Conjectures for the Ashkin--Teller model
  7. About the proof
  8. The discrete coupling
  9. Definition and main properties
  10. The height function
  11. Extension to the Ashkin--Teller model
  12. Properties of the representation
  13. Correlation inequalities
  14. RSW Theory
  15. Construction of the IIC measure
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $D \subset \mathbb C$ be a Jordan domain. Let $h$ be a GFF in $D$ with zero boundary conditions, and let $\xi=(\xi_k)_{k\geqslant 0}$ be i.i.d. symmetric $\{\pm1\}$-valued random variables. Then, there exist (explicit) deterministic measurable functions $\Phi^+$, $\tilde{\Phi}^+$, $\Phi^\textnor

Figures (8)

  • Figure 1: The two-valued set $\mathbb{A}_{-2\lambda, 2\lambda}$, shaded in grey, of a GFF with zero boundary conditions in the domain encircled by some loop, coloured in blue. Every cluster in the decompositions of Theorem \ref{['thm:main-decomp']} is of this form.
  • Figure 2: Some of the clusters under $+$ boundary conditions. The left-hand depicts the boundary cluster $\mathcal{C}_0^+=\mathbb{A}_{-2\lambda, 2\lambda}(h)$, shaded in grey. On the right-hand side, step (2.$+$) has been additionally performed inside only one of the loops $\gamma\in\mathcal{L}_{-2\lambda, 2\lambda}(h)$. Every $\ell\in\mathcal{L}_{-2\lambda, (2\sqrt{2}-2)\lambda}(h^\gamma)$, coloured in blue, is associated a unique cluster, shaded in grey, which has $\ell$ as its outer boundary.
  • Figure 3: Some of the clusters under free boundary conditions. Only step (1.f) is depicted. Every loop $\ell\in\mathcal{L}_{-\sqrt{2}\lambda, \sqrt{2}\lambda}$, coloured blue, is associated a cluster, shaded in grey.
  • Figure 4: A simulation of the boundary cluster of the percolation $\omega$ on a 1000x1000 grid. The law of this boundary cluster converges to that of the carpet of $\textnormal{CLE}_4$.
  • Figure 5: The dashed line is the boundary of a simply connected domain $U$. It defines a domain $U^\mathcal{C}$, with two connected components, by removing all the loops of $\mathcal{C}$ that intersect both the inside and the outside of $U$. The Markov property of the $\textnormal{CLE}_4$ says that the blue loops have the law of (independent) $\textnormal{CLE}_4$ in each of the connected components of $U^\mathcal{C}$.
  • ...and 3 more figures

Theorems & Definitions (117)

  • Theorem 1.1: Four IMFs from one GFF and independent coin tosses
  • Theorem 1.2: Representation of IMFs via two-valued sets of the GFF
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Identification of the measures
  • Definition 1.7: Coin tosses
  • Definition 1.8
  • Theorem 1.9: Alternative Edwards--Sokal coupling
  • Theorem 1.10
  • ...and 107 more