From Diamond to Astroid: How Tilings Took a New Shape
11 May 2026, Lynn
A unified formula now solves domino tiling problems for any periodic lattice, expanding from the Aztec diamond to astroidal shapes.
The Aztec diamond is a legendary shape in statistical physics. It’s not a real diamond — it’s a pattern of squares that you can tile with dominoes. For decades, it has been one of the few examples where physicists and mathematicians could write down exact formulas for how the tilings behave. But what if the Aztec diamond was just the simplest member of an enormous family? A new preprint (arXiv:2605.03896) proposes exactly that.
The work comes from Tomas Berggren, Alexei Borodin, and Terrence George. They have found a way to construct entire families of tiling problems — one for any periodic pattern of interactions on a planar lattice — and solve them exactly. The key lies in a geometric object called the Newton polygon.
What Is a Dimer Model, Really?
Think of a chessboard, but instead of playing chess, you cover it with dominoes. Each domino covers two adjacent squares. That’s a dimer cover. The dimer model is a probability measure on all possible domino tilings, where each tiling gets a weight proportional to the product of the edges used. For a large board, the number of tilings grows exponentially, and the interesting question is: how do dominoes correlate?
That correlation is encoded in a matrix called the Kasteleyn matrix. Inverting it gives you all the correlation functions — the probability that two given squares are covered by the same domino, and so on. For a finite board, the Kasteleyn matrix is just a finite matrix. But for a large board taken from a periodic lattice, computing its inverse explicitly has been a rare art. Before this work, the only known examples had Newton polygons with three or four sides — triangles and quadrilaterals. The Aztec diamond, with its famous circular arctic boundary, is the most celebrated four-sided case.
The Newton Polygon: The DNA of a Lattice
The Newton polygon is a convex lattice polygon that classifies periodic planar bipartite graphs up to local moves. It’s like the graph’s fingerprint. Its number of sides — call it n — determines the complexity of the graph’s tiling behavior. For a square lattice, the Newton polygon is a square (n=4). For a honeycomb lattice, it’s a hexagon (n=6). But until now, no one had constructed finite subgraphs with more than four sides where the Kasteleyn matrix could be inverted in closed form.
Berggren, Borodin, and George realized that for any n, there is a natural family of finite subgraphs whose boundaries are formed by zig-zag paths — paths that turn as sharply as possible at each step, turning maximally right at white vertices and maximally left at black ones. The overall shape of these subgraphs resembles an astroid, the curve traced by a point on a rolling circle. They call them astroidal zig-zag graphs, or AZ graphs. The Aztec diamond is the simplest: when the Newton polygon is a square, the AZ graph reduces to the classic diamond. But now the construction works for pentagons, hexagons, decagons — any polygon you like.
The Master Key: A Double Contour Integral
The heart of the paper is a formula for the inverse Kasteleyn matrix of any AZ graph. It is given by a double contour integral on a curve called the spectral curve. This may sound abstract, but the key point is that it works for any periodic weighting of the graph — including the standard uniform weighting, but also more exotic ones where edges have different probabilities.
Like a master key that opens every lock in a hotel, their formula fits every periodic weighting of any astroidal zig-zag graph. For the Aztec diamond, the formula reproduces the known results. For a decagonal Newton polygon — a ten-sided shape — it produces an explicit expression that no one had before.
Unlike dinner guests who can simultaneously occupy multiple seats, each dimer can only occupy one pair of vertices at a time. The inverse Kasteleyn matrix, expressed as a double contour integral over a spectral curve, encodes the correlations by integrating over all possible Fourier modes of the periodic lattice.
The Arctic Curve Emerges
When the AZ graph grows large, the dimer model undergoes a phase separation. The board splits into three distinct types of regions: a frozen region where dominoes are locked in place, a rough (liquid) region where they fluctuate wildly, and a smooth (gaseous) region in between. The boundary between frozen and rough is called the arctic curve. For the Aztec diamond, it’s a perfect circle. For a general Newton polygon, the arctic curve becomes something richer — a curve that can be computed explicitly from the double integral formula.
The team also derived the limit shape — the deterministic profile of the height function that describes how the tiling slopes. And they proved that local dimer correlations converge to the translation-invariant Gibbs measure determined by that slope. This is the gold standard for large-scale behavior in statistical mechanics.
What This Means
This is not just a technical advance. It reshapes our understanding of what is exactly solvable in statistical physics. For years, the Aztec diamond and a handful of other examples stood as isolated islands of exactness. Now, a whole archipelago appears, connected by the same mathematical machinery. The Newton polygon is the map of that archipelago.
The work is purely theoretical. The formulas involve contour integrals on a spectral curve, which may be hard to evaluate for very complex polygons. But the existence of a unified formula marks a milestone. The next step is to use these exact solutions to probe the fine details of the phase transition — the scaling limit of the arctic curve, the fluctuations of the height function near the boundary.
The road ahead is clear: these AZ graphs provide a controlled laboratory for testing universal predictions about random surfaces and phase boundaries. And because the dimer model maps to other systems — like the six-vertex model, or even certain quantum spin chains — the results may echo far beyond domino tilings.
The Aztec diamond has been the crown jewel for three decades. Now we have a necklace of jewels, each as precious as the first. The field of integrable probability just got a lot richer — and a lot more interesting.
Lynn is an online editor of LoomSci
References
- Tomas Berggren et al., Dimer models on astroidal zig-zag graphs, arXiv:2605.03896