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Massive holomorphicity of near-critical dimers and sine-Gordon model

Nathanaël Berestycki, Scott Mason, Lucas Rey

Abstract

In this paper, we consider the near-critical dimer model in the setup of isoradial superpositions with Temperleyan boundary conditions. We show that the centered height function converges as the mesh size tends to zero to a limiting field which agrees with the (electromagnetically tilted) sine-Gordon model, whose derivative correlations are described by Grassmann variables (or equivalently determinants involving a massive Dirac operator). This answers a longstanding question in the field. A crucial part of the work is to develop a notion of discrete massive holomorphic functions and the tools to study such functions, in particular finding an exact discrete form of the massive Cauchy--Riemann equations, which is satisfied by the inverse Kasteleyn matrix. In comparison with previous studies, a key novelty of this part of our work is that the mass is not only allowed to be non-constant but can be complex-valued.

Massive holomorphicity of near-critical dimers and sine-Gordon model

Abstract

In this paper, we consider the near-critical dimer model in the setup of isoradial superpositions with Temperleyan boundary conditions. We show that the centered height function converges as the mesh size tends to zero to a limiting field which agrees with the (electromagnetically tilted) sine-Gordon model, whose derivative correlations are described by Grassmann variables (or equivalently determinants involving a massive Dirac operator). This answers a longstanding question in the field. A crucial part of the work is to develop a notion of discrete massive holomorphic functions and the tools to study such functions, in particular finding an exact discrete form of the massive Cauchy--Riemann equations, which is satisfied by the inverse Kasteleyn matrix. In comparison with previous studies, a key novelty of this part of our work is that the mass is not only allowed to be non-constant but can be complex-valued.
Paper Structure (41 sections, 39 theorems, 431 equations, 4 figures)

This paper contains 41 sections, 39 theorems, 431 equations, 4 figures.

Table of Contents

  1. Introduction; main results
  2. Background and overview
  3. Heuristics and connection to imaginary geometry.
  4. Discrete and continuous massive holomorphicity.
  5. Isoradial setup; discrete massive harmonic functions
  6. Edge weights.
  7. Interpretation as drift.
  8. Notion of massive holomorphicity in the continuum
  9. Scaling limit for the inverse Kasteleyn matrix
  10. Scaling limit for moments of height function
  11. Identification with sine-Gordon model
  12. Acknowledgements.
  13. Analysis of massive discrete differentials
  14. Discrete massive Laplacian
  15. Gauge equivalence, Kasteleyn matrix.
  16. ...and 26 more sections

Key Result

Theorem 1.1

Suppose $\Omega$ is a bounded simply connected domain. Let $h_\varepsilon$ denote the centered height function associated to the dimer model on the Temperleyan graph $G_\varepsilon \subset \varepsilon \mathbb{Z}^2 \cap \Omega$, associated with weights as in eq:weightsZ2. Suppose that the potential $

Figures (4)

  • Figure 1: Edge weights. (The weights indicated on this picture correspond to the orientation of a given edge from the black vertex towards the white vertex in the middle of the rhombus).
  • Figure 2: Gauge change. An example of Kasteleyn phases is given in red.
  • Figure 3: Straight boundary
  • Figure 4: A simply connected subgraph $\Lambda$ of $\Gamma^{\infty}$, in black. Its restricted dual $\Lambda^{\star}$ in grey. The polylines $x_0w_0\dots x_nw_n$ and $y_0\tilde{w}_0\dots y_m \tilde{w}_m$ are respectively the outer thick red line and inner thick blue line.

Theorems & Definitions (95)

  • Theorem 1.1
  • Remark 1.3
  • Lemma 1.4
  • Definition 1.6
  • Remark 1.7
  • Lemma 1.8
  • proof
  • Remark 1.9
  • Definition 1.10
  • Definition 1.11
  • ...and 85 more