Split Two-Periodic Aztec Diamond

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.

💡 Research Summary

The paper introduces a novel variant of the Aztec diamond dimer model in which the usual two‑periodic weighting is split along a vertical line, creating a “split two‑periodic” Aztec diamond. On the left half of the diamond the edge weights follow the standard two‑periodic pattern with parameters (α, β), while on the right half the pattern is reversed to (β, α). The interface at x = 2N (with N = n/2) therefore breaks periodicity in one direction while preserving the determinantal structure of the model.

To analyse this non‑uniform system the authors extend the Wiener‑Hopf factorisation technique developed by Berggren and Duits (2019) and combine it with the Eynard‑Mehta theorem, which connects the dimer model to a non‑intersecting paths process (the BD‑paths graph). They first define a family of 2 × 2 matrix‑valued functions Φε(z) (ε = α or β depending on the row index) and obtain an explicit eigen‑decomposition Φε(z)=Eε(z) diag(rε,1(z), rε,2(z)) Eε(z)⁻¹. The eigenvalues rε,1, rε,2 are rational functions of z involving the parameters α, β, and the associated eigenvectors give rise to projection matrices Fε,1(z) and Fε,2(z).

A new scalar function gα,β(z) is introduced to capture the interaction between the two weightings; it is symmetric in α and β and possesses branch cuts that separate the complex plane into regions associated with the left and right halves of the diamond.

The main result (Theorem 2.1) expresses the correlation kernel K_N for the split model as a double contour integral. The outer integral runs over a small circle γ₀,₁ around the origin and a circle γ₁ around 1, while the inner integral runs over the same γ₁. Depending on whether the row index m′ lies in the lower half (0 < m′ ≤ N/2) or the upper half (N/2 < m′ < N), the kernel involves Φα or Φβ, respectively, together with the projection matrices and the factor (1 + 2 gα,β(w)). When m ≤ N/2 the parameter ε in Φε is set to α, otherwise to β. This piecewise structure reflects the change of weighting across the interface.

Having obtained a kernel suitable for asymptotic analysis, the authors classify the macroscopic phases of the model into frozen, rough, and smooth regions, exactly as in the classical two‑periodic Aztec diamond. However, the presence of the interface creates additional sub‑regions where the smooth‑rough boundary meets the interface. Using saddle‑point methods they deform the contours and evaluate the integrals Iα,2,k, leading to explicit local formulas. The leading order asymptotics coincide with those of the ordinary two‑periodic model, confirming that the global shape (the arctic curve) is unchanged. The novelty appears in the sub‑leading terms: the factor gα,β(w) modifies the decay rate of correlations in the smooth region, producing an anisotropic correction that depends on the distance to the interface and on the difference β − α. In particular, near the point where the smooth‑rough boundary intersects the interface, the correlation decay exhibits a new cusp‑like behavior not present in the uniform two‑periodic case.

The paper concludes by outlining how the same Wiener‑Hopf framework can be adapted to other non‑periodic or partially periodic weightings, such as q‑vol weights or Fock‑type weights, and suggests studying the dynamics of the interface itself (e.g., moving it left or right, or introducing multiple interfaces). The results provide a concrete prescription for deriving correlation kernels in a broad class of non‑periodic dimer models and highlight how sub‑leading asymptotics can reveal subtle new phenomena even when the leading macroscopic picture remains unchanged.