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Marked GUE-corners process in doubly periodic dimer models

Tomas Berggren, Nedialko Bradinoff

Abstract

We study a family of periodically weighted Aztec diamond dimer models near their turning points. We establish that, asymptotically, as $N\rightarrow\infty$, their fluctuations there, scaled by $\sqrt{N}$, are described by a marked GUE-corners process. This limiting point process is constructed by assigning a Bernoulli mark independently to each particle in a realization of the GUE-corners process. The Bernoulli parameters associated with the random marks reflect the periodicity of the model in the limit. To prove this result we use a double-contour integral representation of the inverse Kasteleyn matrix on a higher-genus Riemann surface, which is well-suited for asymptotic analysis.

Marked GUE-corners process in doubly periodic dimer models

Abstract

We study a family of periodically weighted Aztec diamond dimer models near their turning points. We establish that, asymptotically, as , their fluctuations there, scaled by , are described by a marked GUE-corners process. This limiting point process is constructed by assigning a Bernoulli mark independently to each particle in a realization of the GUE-corners process. The Bernoulli parameters associated with the random marks reflect the periodicity of the model in the limit. To prove this result we use a double-contour integral representation of the inverse Kasteleyn matrix on a higher-genus Riemann surface, which is well-suited for asymptotic analysis.

Paper Structure

This paper contains 21 sections, 25 theorems, 130 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preface
  3. The Aztec diamond and an interlacing particle system
  4. The GUE-corners process
  5. The marked GUE-corners process
  6. Main results
  7. Outlook on further developments
  8. Outline of the paper.
  9. Acknowledgements
  10. Preliminaries
  11. The Aztec diamond dimer model
  12. An interlacing particle system
  13. The correlation kernel and the spectral curve
  14. Convergence to the marked GUE-corners process
  15. Asymptotic analysis of the correlation Kernel
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $K_{\operatorname{Int}}$ be the correlation kernel of $(u_{s}^t)_{1\leq s\leq t\leq 2\ell N}$ and suppose $\nu:\mathbb{Z}_{>0}\times \{0,1\} \rightarrow \mathbb{R}$ is given by where $\alpha_{\ell-t}$ and $\beta_{\ell-t}$ are the edge weights of the model (see Figure fig:Aztec_diamond_graph_4 and later eqn:edge_weights). Let the gauge function $g:\{0,1,\ldots, 2\ell N\}^2\rightarrow \mathbb{R

Figures (7)

  • Figure 1: The Aztec diamond graph of size 4 on the left and the fundamental domain of the $2\times 4$ periodically weighted Aztec diamond dimer model on the right. North, East, South, and West edges are colored blue, orange, red, and green respectively.
  • Figure 2: The two interlacing particle systems near the turning point, in grey and black, defined from South (yellow) and West (red) edges from a random outcome of the Aztec diamond; The picture is rotated so that the first particle is at the bottom instead of to the right. Dimers in the picture are illustrated as dominoes.
  • Figure 3: Left: The particles defined from the dimers visualized in red and cyan. The color depends on the parity of the coordinates of the particle. Right: A dimer cover of the Aztec diamond of size $4$. The particles form an interlacing particle system.
  • Figure 4: The Riemann surface $\mathcal{R}$ represented as two copies of the complex plane. The cuts (dashed) are located along the negative part of the real lines. The compact ovals $A_k$, $k=1,\dots,\ell-1$ (solid) are located along the negative part of the real lines, and the non-compact oval $A_0$ (solid) are located along the positive part of the real lines. The gray areas are connected via the cuts, and so are the white areas. In this illustration, the curve $\tilde{\Gamma}_s$ (blue) is a simple loop, while $\tilde{\Gamma}_l$ (red) is the union of two simple loops.
  • Figure 5: Contours of integration for the integral representation of the GUE-corners process; $\gamma_s'$ is drawn in blue and $\gamma_{\ell'}$ is drawn in red when $\mu_1<\mu_2$ and in a dashed grey line for $\mu_1> \mu_2$.
  • ...and 2 more figures

Theorems & Definitions (56)

  • Theorem 1.1: Theorem \ref{['thm:limit_correlation_function']}
  • Theorem 1.2: Corollary \ref{['cor:week_converence']}
  • Corollary 1.3: Corollary \ref{['cor:thinned']}
  • Lemma 2.1
  • proof
  • Lemma 2.2: JN06
  • proof
  • Lemma 2.3: Ber21
  • Theorem 2.4
  • proof
  • ...and 46 more