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aztec

domino tilings · the arctic circle theorem · 1992–1995

↑ geometry pack · loop-family siblings: meander · temperley–lieb

The Aztec diamond of order n is a staircase diamond of 2n(n+1) squares; it can be tiled by 2×1 dominoes in 2n(n+1)/2 ways. Pick one uniformly at random (here, by domino shuffling) and something astonishing happens: the corners freeze into four solid blocks of one domino type each, the centre stays a roiling jumble — and the boundary between them converges to a perfect circle. That's the Arctic Circle theorem. Resample, or grow the diamond order by order and watch the circle sharpen. Then drag the bias: tilt the creation coin toward horizontal dominoes and the circle stretches into the arctic ellipse X²/p + Y²/(1−p) = 1 — same algorithm, one knob turned.

order · n
32
creation bias · p (horizontal)
0.50
Each colour is one of the four domino types — N / S / E / W — and the theorem says each owns one frozen corner. At bias p = ½ the temperate region is the inscribed circle (area π/4). Off ½ it's the arctic ellipse with semi-axes √p and √(1−p) (so the axes-squared always sum to 1); its area fraction is π√(p(1−p))/2, peaking at p = ½. More horizontal creations ⇒ a wider, flatter liquid lens and fatter E/W frozen corners.

The shuffle grows the diamond by one ring at a time, in three moves. Press next phase and watch a tiny diamond do it, step by step — the only randomness is a coin flip per new hole, yet it keeps every tiling equally likely.

order
current n1
Destroy: delete every 2×2 brick whose two dominoes point into each other (they'd collide). Slide: each survivor steps one cell the way it faces (N↑ S↓ E→ W←). Create: every empty 2×2 hole gets two dominoes — a coin flip picks horizontal or vertical.

The Aztec diamond

The Aztec diamond ADn is the union of the unit squares whose centres (x+½, y+½) satisfy |x+½| + |y+½| ≤ n — a diamond with staircase sides, 2n(n+1) squares in all. Elkies, Kuperberg, Larsen and Propp proved in 1992 that the number of ways to tile it with 2×1 dominoes is exactly 2n(n+1)/2 — a clean power of two, which is already a small miracle.

The Arctic Circle theorem

Now take a tiling uniformly at random. Jockusch, Propp and Shor (1995) proved that as n → ∞, the tiling splits into two phases separated by a sharp curve that, rescaled, is exactly the circle inscribed in the diamond — radius n/√2:

A purely combinatorial object — dominoes on a board — spontaneously grows a perfect circle. It is one of the cleanest limit-shape phenomena in mathematics, and the founding example of an industry (limit shapes of random surfaces, the Ising and dimer models, the Kardar–Parisi–Zhang universality class).

No one drew the circle. It is what a random tiling looks like.

Domino shuffling

How do you sample a tiling uniformly — each of 2n(n+1)/2 equally likely — without listing them all? The shuffling algorithm of Elkies–Kuperberg–Larsen–Propp builds ADn+1 from a uniform ADn in three moves:

  1. Destruction. Find every 2×2 block holding two dominoes that point into each other and delete them.
  2. Sliding. Every surviving domino slides one step in its type-direction (N up, S down, E right, W left).
  3. Creation. The empty 2×2 blocks that open up each get a fresh pair of dominoes — two horizontal or two vertical — by a fair coin flip.

Each coin flip is the only randomness, and the magic is that this exactly preserves uniformity. The grow +1 button runs one step of this on the live tiling; resample runs it from order 1 with fresh coins. (The implementation here was checked against brute-force enumeration: it reproduces the 2, 8, 64, 1024 tiling counts of AD₁–₄ and samples them uniformly.)

Biasing the coin: the arctic ellipse

The shuffle has exactly one source of randomness — the coin flipped to fill each fresh 2×2 hole. Make it unfair. Let p be the probability the hole is filled with two horizontal dominoes (and 1−p vertical). This is the weighted shuffle (Propp's generalized domino-shuffling): with a spatially uniform bias the destruction and sliding carry the weights through unchanged, so a single global p samples the weighted tiling measure. Cohn, Elkies & Propp's local statistics paper (math/0008243) is exactly the apparatus that pins down what such a biased ensemble looks like in the large.

The answer is clean: the arctic circle becomes an arctic ellipse. In rescaled coordinates X = (x+½)/n, Y = (y+½)/n (the diamond is |X|+|Y| ≤ 1), the frozen/temperate boundary converges to

X²/p + Y²/(1−p) = 1  — semi-axes √p and √(1−p).

A few things fall right out. The semi-axes squared always sum to one (√p² + √(1−p)² = 1), so the ellipse breathes between a horizontal sliver and a vertical one while staying snug inside the diamond, hitting the circle exactly at p = ½. The temperate area fraction is π√(p(1−p))/2, maximal (π/4 ≈ 0.785) at p = ½ and shrinking as you push the bias either way — order is cheaper when the dominoes have a preferred direction. Both the ellipse and the area law were checked here against the sampler (semi-axis √(1−p) and the area fraction match the simulated boundary to within finite-size error). Drag the bias on the sample tab and watch the dashed curve — and the four corners — deform.

Why it sits in the geometry pack

Domino tilings are the dimer model — perfect matchings of a lattice — a first cousin of the loop model behind Temperley–Lieb and the non-crossing world of meanders. The same statistical-mechanics machinery (height functions, Kasteleyn's determinant, conformal field theory) runs underneath all three. The arctic circle is the loud, visual face of it: poke a simple object, and a limit shape falls out.

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