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Split Two-Periodic Aztec Diamond

Meredith Shea

TL;DR

This work introduces the Split Two-Periodic Aztec Diamond, a non-periodic extension of the classic two-periodic dimer model with an interface that breaks periodicity along a line. By extending Berggren–Duits' framework and applying the Eynard-Mehta theorem together with Wiener–Hopf factorization, the authors derive a detailed integral representation for the correlation kernel and relate it to the standard two-periodic kernel away from the interface. They classify macroscopic regions via saddle-function analysis and provide precise local asymptotics for the kernel, showing that subleading terms differ from the usual model near the interface while leading-order behavior matches the two-periodic case. The work also develops an intermediate kernel through a non-intersecting paths formulation, enabling a structured path to general non-periodic weightings and offering insights into how interface-induced coupling alters decay rates in the smooth region. Overall, the results advance understanding of non-periodic dimer models and supply a robust methodology for deriving kernels in similar non-periodic settings with interfaces.

Abstract

Recent advancements have been made to understand the statistics of the Aztec diamond dimer model under general periodic weights. In this work we define a model that breaks periodicity in one direction by combining two different two-periodic weightings. We compute the correlation kernel for this Aztec diamond dimer model by extending the methods developed by Berggren and Duits (2019), which utilize the Eynard-Mehta theorem and a Wiener-Hopf factorization. From a form of the correlation kernel that is suitable for asymptotics, we compute the local asymptotics of the model in the different macroscopic regions present. We prove that the local asymptotics of the model agree with the typical two-periodic model in the highest order, however the sub-leading order terms are affected.

Split Two-Periodic Aztec Diamond

TL;DR

This work introduces the Split Two-Periodic Aztec Diamond, a non-periodic extension of the classic two-periodic dimer model with an interface that breaks periodicity along a line. By extending Berggren–Duits' framework and applying the Eynard-Mehta theorem together with Wiener–Hopf factorization, the authors derive a detailed integral representation for the correlation kernel and relate it to the standard two-periodic kernel away from the interface. They classify macroscopic regions via saddle-function analysis and provide precise local asymptotics for the kernel, showing that subleading terms differ from the usual model near the interface while leading-order behavior matches the two-periodic case. The work also develops an intermediate kernel through a non-intersecting paths formulation, enabling a structured path to general non-periodic weightings and offering insights into how interface-induced coupling alters decay rates in the smooth region. Overall, the results advance understanding of non-periodic dimer models and supply a robust methodology for deriving kernels in similar non-periodic settings with interfaces.

Abstract

Recent advancements have been made to understand the statistics of the Aztec diamond dimer model under general periodic weights. In this work we define a model that breaks periodicity in one direction by combining two different two-periodic weightings. We compute the correlation kernel for this Aztec diamond dimer model by extending the methods developed by Berggren and Duits (2019), which utilize the Eynard-Mehta theorem and a Wiener-Hopf factorization. From a form of the correlation kernel that is suitable for asymptotics, we compute the local asymptotics of the model in the different macroscopic regions present. We prove that the local asymptotics of the model agree with the typical two-periodic model in the highest order, however the sub-leading order terms are affected.

Paper Structure

This paper contains 35 sections, 19 theorems, 208 equations, 19 figures.

Table of Contents

  1. Introduction
  2. The Aztec Diamond and Dimer Models
  3. Acknowledgements
  4. The Split Two-Periodic Aztec Diamond
  5. A Related Paths Process and the Correlation Kernel
  6. Summary of Results
  7. An Integral Representation of the Correlation Kernel
  8. Comparison to the Two-Periodic Aztec Diamond
  9. Classification of the Macroscopic Regions
  10. Asymptotic Behavior
  11. Derivation of the Correlation Kernel
  12. An Intermediate Correlation Kernel
  13. Proof of Theorem \ref{['theorem:kernel']} when $0 < m' \leq N/2$
  14. Proof of Theorem \ref{['theorem:kernel']} when $N/2< m' < N$
  15. Local Asymptotics of the Model
  16. ...and 20 more sections

Key Result

Theorem 2.1

Let $-N \leq \xi,\xi' \leq -1$ and $0 < m < N$. The split two-periodic Aztec diamond of size $2N$ has correlation kernel given by when $0 < m' \leq N/2$ and when $N/2 < m' <N$. For both cases, if $m \leq N/2$ then $\varepsilon = \alpha$ and if $m > N/2$ then $\varepsilon = \beta$. Additionally, $\gamma_1$ is a contour surrounding $1$ and not $0$, while $\gamma_{0,1}$ is a contour surrounding $0$

Figures (19)

  • Figure 1: On the left is a simulation of the Aztec diamond with uniform weights. On the right is a simulation of the Aztec diamond with two periodic weights. Original code for simulation was provided by Sunil Chhita.
  • Figure 2: Simulated split Aztec diamond where $N = 200$, $\alpha = 1/4$, and $\beta = 1/2$. The tiles are colored by an 8 color gray scale to accentuate the smooth region. Original code for simulation was provided by Sunil Chhita. Simulations are sampled exactly from the distribution using domino-shuffling, see Pro03.
  • Figure 3: Examples of the Aztec diamond graph. On the left is the Aztec diamond of size $n = 3$, on the right is the Aztec diamond of size $n = 4$.
  • Figure 4: The Aztec diamond of size $2N = 4$ with coordinates given by equations \ref{['eq:coordstart']}-\ref{['eq:coordend']} and weights given by equation \ref{['eq:edgewts']}. All unlabeled edged have weight 1.
  • Figure 5: An example of the BD-paths graph for $n = 4$. The graph is also equipped with the edge weights that make it equivalent to the split two-periodic Aztec diamond, see Section \ref{['section:BDstuff']} for more details. Unlabeled edges have weight 1.
  • ...and 14 more figures

Theorems & Definitions (36)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • Remark 2.1
  • Theorem : BD19, Theorem 5.2
  • Remark 2.2
  • Remark 2.3
  • Definition 2.1
  • Remark 2.4
  • Proposition 2.1
  • ...and 26 more