The t-Split Two-Periodic Aztec Diamond Model

Technical Summary: The t-Split Two-Periodic Aztec Diamond Model

Problem Statement
This work investigates a generalization of the Aztec diamond dimer model, specifically extending the "Split Two-Periodic Aztec Diamond" analyzed in prior work [16]. The standard Aztec diamond is a subset of the square grid $\mathbb{Z}^2$ defined by $|x| + |y| \leq n$. The model under consideration is a dimer model (perfect matchings) on this graph, where edges are assigned weights that vary periodically.

The specific innovation of this paper is the introduction of an asymmetric interface. The Aztec diamond of size $n=2N$ is split into two regions by a vertical line at $x = 4tN$, where $0 < t < 1/2$. The region to the left of the interface ($x < 4tN$) is assigned a two-periodic weighting scheme governed by parameter $\alpha$, while the region to the right ($x > 4tN$) is assigned a distinct two-periodic weighting governed by parameter $\beta$. This differs from the previous "Split" model [16] where the interface was fixed at the center ($t=1/2$), creating two equal regions. The paper refers to the right side as the "dominant" side and the left side as the "non-dominant" side.

Methodology
The analysis relies on the framework of determinantal point processes and the asymptotic analysis of correlation kernels.

1. Coordinate Transformation: The authors define a coordinate transformation on the Aztec diamond vertices to align the model with the non-intersecting paths process convention used in Berggren and Duits [3]. This involves shifting white vertices and rescaling the $y$-coordinates.
2. Kernel Derivation: The primary technical tool is the derivation of the correlation kernel $K_N$. The authors utilize the Wiener-Hopf factorization method. To handle singularities on the unit circle inherent in the two-periodic weights, they introduce a bias parameter $a \in (0,1)$ into the weight matrices $\Phi_{\varepsilon, a}(z)$, perform the factorization, and then rigorously take the limit $a \to 1$.
   • The derivation involves iterative matrix identities to eliminate poles that converge to the same point as $a \to 1$.
   • The resulting kernel is expressed as a piecewise function depending on the position of the coordinates relative to the interface ($m \leq tN$ vs. $m > tN$).
3. Asymptotic Analysis: The scaling limit is analyzed using the method of steepest descent applied to the integral representation of the kernel. The macroscopic regions (frozen, rough, smooth) are determined by the behavior of saddle points of a "saddle function" defined on a Riemann surface.

Key Contributions and Results

• Theorem 1 (Correlation Kernel): The paper derives an explicit integral expression for the correlation kernel of the $t$-split model. The kernel is a $2 \times 2$ matrix involving double contour integrals. It recovers the known two-periodic kernel when $\alpha = \beta$ and the split kernel when $t=1/2$.
• Proposition 1 (Dominant Side Asymptotics): For the region to the right of the interface ($m, m' > tN$), the authors prove that the correlation kernel converges to that of a standard two-periodic Aztec diamond with parameter $\beta$. The error term decays exponentially ($O(e^{-cN})$). Consequently, the limit shape on the dominant side is identical to the limit shape of the standard two-periodic model with parameter $\beta$, independent of $\alpha$ or the interface location $t$.
• Conjecture 1 (Non-Dominant Side Asymptotics): For the region to the left of the interface ($m, m' < tN$), the authors propose a conjecture for the limit shape. They define a new saddle function $\phi_{\alpha, \beta}(z; x, y)$ which depends on both parameters $\alpha, \beta$ and the interface location $t$. The macroscopic boundary is conjectured to be determined by the coalescing of the critical points of this function. Unlike the dominant side, the limit shape here depends on all three parameters.
• Conjecture 2 (Existence of Smooth Region): Based on numerical evidence, the paper conjectures that a smooth region exists to the left of the interface if and only if $\frac{\beta^2}{1+\beta^2} < t$. This suggests that the existence of the smooth region on the non-dominant side is governed by the parameters of the dominant side ($\beta$) and the interface location ($t$), rather than the local parameter $\alpha$.
• Proposition 2 (Special Case $\beta = 0$): The paper provides a complete proof for the limit shape in the case where the dominant side parameter $\beta = 0$. In this limit, the rough region vanishes on the dominant side. The authors prove that the limit shape on the non-dominant side becomes exactly half of a rescaled two-periodic Aztec diamond (with parameter $\alpha$), restricted to the left half.

Significance
The paper establishes that the local asymptotic behavior of the split model is not always determined solely by the local weighting. While the dominant side behaves exactly as a standard two-periodic model, the non-dominant side exhibits a "novel" limit shape that is influenced by the global parameters of the system (the weighting of the other side and the interface position).

The work provides a rigorous integral formula for the correlation kernel, which is a prerequisite for further asymptotic analysis. While the full asymptotic proof for the non-dominant side is not completed due to the complexity of the critical points of the proposed saddle function, the paper offers a concrete conjecture and numerical validation for the behavior of the model. This generalizes previous results on symmetric split models and offers insight into how interfaces in dimer models with periodic weights affect macroscopic limit shapes.