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Arctic curves of periodic dimer models and generalized discriminants

Mateusz Piorkowski

Abstract

We compute the algebraic equation for arctic curves of the Aztec diamond with a doubly (quasi-)periodic weight structure and obtain similar results for certain models of the hexagon. In particular, we determine the algebraic degree of such curves as a function of the number of frozen and smooth (or gaseous) regions. The key to our result is the construction of a discriminant for meromorphic differentials on a higher genus Riemann surface. This construction works analogously for meromorphic sections of arbitrary holomorphic line bundles. In the genus $g = 0$ case this notion reduces to the usual discriminant of a polynomial.

Arctic curves of periodic dimer models and generalized discriminants

Abstract

We compute the algebraic equation for arctic curves of the Aztec diamond with a doubly (quasi-)periodic weight structure and obtain similar results for certain models of the hexagon. In particular, we determine the algebraic degree of such curves as a function of the number of frozen and smooth (or gaseous) regions. The key to our result is the construction of a discriminant for meromorphic differentials on a higher genus Riemann surface. This construction works analogously for meromorphic sections of arbitrary holomorphic line bundles. In the genus case this notion reduces to the usual discriminant of a polynomial.

Paper Structure

This paper contains 19 sections, 12 theorems, 87 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of the paper
  4. (Quasi-)periodic dimer models
  5. -periodic Aztec diamond
  6. Other dimer models
  7. Notion of a discriminant on a compact Riemann surface
  8. Definition of -discriminants
  9. Construction of -discriminants
  10. Genus case
  11. Higher genus case
  12. Generalizations
  13. Results on arctic curves
  14. Irreducibility of -discriminants related to dimer models
  15. Arctic curves of the Aztec diamond
  16. ...and 4 more sections

Key Result

Lemma 2.1

(simplified version, cf. BB23+) A point $(x_1, x_2) \in [-1,1]^2$ lies on the arctic curve if and only if the meromorphic differential $dF$ has a zero of higher multiplicity.

Figures (12)

  • Figure 1: A random tiling of the Aztec diamond of size $4$ (left) and of size $200$ (right). Note the frozen regions in the corners and the emergence of a disc-shaped rough region in the middle. In this case the arctic curve becomes a circle as proven by Jockusch, Propp and Shor JPS (generated using code kindly provided by Christophe Charlier).
  • Figure 2: Left: Large tiling of the $2 \times 2$-periodic model of the Aztec diamond studied in CJ16, CY14, DK21 with one smooth and four frozen regions (size $300$); Right: The arctic curve of degree eight for the corresponding model (left image generated using code kindly provided by Christophe Charlier).
  • Figure 3: Graph of the $\boldsymbol{\eta}$-discriminant $\Delta_{\boldsymbol{\eta}}$ of the $2\times2$-periodic model from Fig. \ref{['LargeTiling']}. Its zero set highlighted in black coincides with the arctic curve of the model.
  • Figure 4: Aztec diamond graph of size 4.
  • Figure 5: Map between dimers and dominos. The use of similar colors is necessary for the smooth region to be distinguishable from the rough region.
  • ...and 7 more figures

Theorems & Definitions (34)

  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.5
  • Definition 3.1
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6: $\boldsymbol{\eta}$-discriminant
  • ...and 24 more