Wiener-Hopf factorizations and matrix-valued orthogonal polynomials

TL;DR

The paper analyzes periodic dimer models for the weighted Aztec diamond through two analytical lenses: matrix-valued orthogonal polynomials (MVOPs) and Wiener–Hopf factorizations. It proves the two approaches are equivalent in the Aztec diamond setting and, in the notable 2×2 periodic case, derives explicit MVOPs and Wiener–Hopf factors expressed via Jacobi theta functions on a genus-one spectral curve, using a diagonalization of the weight and a careful construction of E, G, and E_ω. It further extends the double-contour integral framework for correlation kernels to k-periodic Aztec diamonds with relaxed pole–zero separation constraints. The work bridges MVOPs, RH problems, and spectral-data to produce explicit, theta-function–based formulas with potential connections to domino-shuffle dynamics and higher-genus generalizations. Overall, it strengthens the analytic toolkit for periodic dimer models and paves the way for richer exact representations in related tiling problems.

Abstract

We compare two methods for analysing periodic dimer models. These are the matrix-valued orthogonal polynomials approach due to Duits and one of the authors, and the Wiener-Hopf approach due to Berggren and Duits. We establish their equivalence in the special case of the Aztec diamond. Additionally, we provide explicit formulas for the matrix-valued orthogonal polynomials/Wiener-Hopf factors in the case of the $2 \times 2$-periodic Aztec diamond in terms of Jacobi theta functions related to the spectral curve of the model.

Figures (5)

• Figure 1: A dimer configuration and the corresponding tiling of the Aztec diamond of size $4$.
• Figure 2: Weighted Aztec diamond graph of size $3$
• Figure 3: The closed contour $\Gamma$ and the domains $\Omega_{int}$ and $\Omega_{ext}$ for the $k$-periodic Aztec diamond considered in Section \ref{['SectkPer']}.
• Figure 4: The two-sheeted Riemann surface $\mathcal{R}$ with branch cuts along $[x_3,x_2]$ and $[x_1,x_0]$, and with $\textbf{a}$ and $\textbf{b}$ cycles. The Riemann surface has special points $p^*$ and $p^{**}$ on the bounded oval, and $p_1$ and $p_2$ on the unbounded oval. The points $p^*$ and $p^{**}$ are given by \ref{['pstar']} and they could be anywhere on the bounded oval. The points $p_1$ and $p_2$ given by \ref{['polelambda']} are the poles of $\lambda$ and they lie on the positive real axis of the first sheet. The zeros of $\lambda$ (not shown in the figure) are on the positive real axis of the second sheet.
• Figure 5: Image of $\mathcal{R}' = \{ p \in \mathcal{R} \mid z(p) \not\in [x_3,x_1]\}$ under the Abel map \ref{['Abelmap']}. The region in dark (light) gray denotes the image of the upper (lower) sheet. As in Figure \ref{['RSurface']}, the point $p^*$ ($p^{**}$) is assumed to be on the first (second) sheet of the bounded oval. In the figure, the branch points $x_1, x_2, x_3$ are considered to be on the first sheet. So for example $\mathcal{A}_+(x_2)$ denotes the limit of $\mathcal{A}(q)$ as $q$ tends to $x_2$ with $q$ in the upper half plane of the first sheet.

Theorems & Definitions (36)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Theorem 2.6
• Remark 2.7
• proof
• Lemma 3.1
• proof