Scaling-limit theory for interacting k-tilings of the Aztec diamond: arctic curves, color coupling, and edge fluctuations

The interacting k-tilings of the Aztec diamond — k superposed domino tilings weighted by $t^{\#\text{interactions}}$, introduced by Corteel–Gitlin–Keating and samplable exactly via the multi-color shuffling algorithm of Keating–Nicoletti — display arctic curves and limit shapes in simulation, but no rigorous asymptotic analysis exists away from the degenerate couplings $t=0,1,\infty$. This problem asks for that theory: explicit limit shapes and arctic curves at fixed intermediate $t$, an explanation of the numerically observed coincidence of the colors' arctic curves, identification of the conjectured one-parameter deformation of $k$ independent Airy processes governing coupled edge fluctuations, and the color–color structure of global height fluctuations; extending the shuffling algorithm to arbitrary inhomogeneous weights is an auxiliary subgoal implied by the source's stated weight restrictions on its construction.

Background

A domino tiling of the Aztec diamond of rank $N$ is a covering by $1\times 2$ rectangles of the diamond-shaped subregion of the square lattice with $N$ squares along each side. The arctic circle theorem of Jockusch, Propp, and Shor states that a uniformly random such tiling develops, as $N\to\infty$, four frozen (perfectly ordered) corner regions and a disordered temperate center, separated by an algebraic arctic curve — the simplest example of a limit shape in random tilings. They proved it with the domino shuffling algorithm, a local resampling procedure that grows a random tiling of the rank-$N+1$ diamond from one of rank $N$; Nordenstam later showed the same algorithm yields the fine-scale statement that boundary fluctuations, under the KPZ-universality $1{:}2{:}3$ scaling (fluctuations of order $N^{1/3}$ over a correlation length of order $N^{2/3}$ in a system of linear size $N$), converge to the Airy process — the universal stationary process built from the Airy kernel that also governs the spectral edge of large random Hermitian matrices — a result first established for this setting by Johansson.

The interacting k-tiling ensemble of Corteel–Gitlin–Keating superposes $k$ domino tilings of the same rank-$N$ Aztec diamond, colored $1,\dots,k$ and stacked in color order. Where the overlay produces one of two specific local configurations — a smaller-color domino lying entirely underneath a larger-color domino — one counts an interaction, and the ensemble carries the Boltzmann weight $t^{\#\text{interactions}}$; at $k=2$, $t=1$ the two tilings are independent (the double dimer model). Corteel–Gitlin–Keating computed the generating polynomials of these ensembles via colored (quantum-integrable) vertex models and proved arctic curves only in the degenerate limits $t=0$ and $t\to\infty$. Keating–Nicoletti then constructed a multi-color generalization of the domino shuffling algorithm — based on a generalized spider move, a local Yang–Baxter-type resampling of the colored overlay — that samples exactly from the $t$-weighted measure for uniform and certain other special weights. Their simulations exhibit arctic curves and limit shapes for all $t$, and, surprisingly, that at intermediate $t$ the arctic curves of the different colors appear to coincide even though the interaction is not symmetric between colors.

The obstacle is fundamental: for fixed $t\notin\{0,1,\infty\}$ the measure is genuinely interacting (non-free-fermion), meaning it is not determinantal, so the Kasteleyn-matrix and Pfaffian technology — the determinantal representation of dimer-model correlations — that underlies essentially all one-color Aztec asymptotics is unavailable, and the inter-color coupling entangles the layers in a way the one-color shuffling analysis does not address.

Problem Statement

For the interacting k-tiling ensemble on rank-$N$ Aztec diamonds with Boltzmann weight $t^{\#\text{interactions}}$, establish a rigorous scaling-limit theory at fixed $k\ge 2$ and fixed intermediate coupling $t\in(0,\infty)\setminus\{1\}$ as $N\to\infty$, by:

1. Proving the existence of, and explicitly determining, the limit shape and arctic curve of each color.
2. Explaining the numerically observed coincidence of the colors' arctic curves at intermediate $t$ — established so far only in the degenerate limits $t\to0$ and $t\to\infty$ — or proving that the curves genuinely differ.
3. Identifying the edge fluctuation process near the arctic curve, conjectured to be a one-parameter deformation of $k$ independent Airy processes with nontrivially coupled edge fluctuations, and determining the global height fluctuation field together with its color–color coupling.

These components jointly serve one objective: replacing the simulation observations of Keating–Nicoletti with a rigorous asymptotic theory of this ensemble. The auxiliary problem implied by the same source's stated weight restrictions on its shuffling construction — extending the generalized-spider-move shuffling algorithm from uniform and the specific allowed weights to arbitrary inhomogeneous weights — is a method-enabling subgoal, not a substitute for the asymptotic theory.

References

1. David Keating and Matthew Nicoletti, 'Shuffling Algorithm for Coupled Tilings of the Aztec Diamond', Ann. Henri Poincaré 25 (2024) 5187–5229. DOI: 10.1007/s00023-023-01407-w; arXiv:2303.09089. https://arxiv.org/abs/2303.09089
2. Sylvie Corteel, Andrew Gitlin, and David Keating, 'Colored vertex models and k-tilings of the Aztec diamond', Trans. Amer. Math. Soc. 379 (2026) 6013–6058. DOI: 10.1090/tran/9333; arXiv:2202.06020. https://arxiv.org/abs/2202.06020
3. Amol Aggarwal, Alexei Borodin, and Michael Wheeler, 'Coloured corner processes from asymptotics of LLT polynomials', Adv. Math. 451 (2024) 109781. DOI: 10.1016/j.aim.2024.109781; arXiv:2309.05970. https://arxiv.org/abs/2309.05970
4. Philippe Di Francesco and David Keating, 'Limit shapes for Domain-Wall (colored) vertex models', J. Phys. A 59 (2026) 165201. DOI: 10.1088/1751-8121/ae585a; arXiv:2509.25442. https://arxiv.org/abs/2509.25442
5. Jonah Guse, David Jiang, and David Keating, 'Colored Vertex Models and Interacting Reverse Plane Partitions', preprint (2025). arXiv:2505.13806. https://arxiv.org/abs/2505.13806
6. Natasha Blitvić and Leonid Petrov, 'Colored Interlacing Triangles and Genocchi Medians', SIGMA 22 (2026), Paper 081, 30 pp. DOI: 10.3842/SIGMA.2026.081; arXiv:2602.04390. https://arxiv.org/abs/2602.04390
7. William Jockusch, James Propp, and Peter Shor, 'Random Domino Tilings and the Arctic Circle Theorem', preprint (1998). arXiv:math/9801068. https://arxiv.org/abs/math/9801068
8. Kurt Johansson, 'The arctic circle boundary and the Airy process', Ann. Probab. 33 (1) (2005) 1–30. DOI: 10.1214/009117904000000937. https://doi.org/10.1214/009117904000000937
9. Eric Nordenstam, 'On the Shuffling Algorithm for Domino Tilings', Electron. J. Probab. 15 (2010), Paper No. 3, 75–95. DOI: 10.1214/ejp.v15-730. https://doi.org/10.1214/ejp.v15-730